sonometer law of length experiment class 12 observation table

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Sonometer experiment ‒ objective, procedure, and tips | labkafe.

Previously, we have talked about what a sonometer is and why we use it in class 11-12 physics labs all around. Now that you know about the sonometer theory, we will tell you how to do the sonometer experiment today. This is a very important, and sometimes quite difficult experiment for students of classes XI XII of CBSE ICSE IGCSE State Boards. Nevertheless, if you follow the procedure given below and note the insights from our experts, you will be able to do it with good results.

A sonometer is one of the most important lab equipment in most high school level physics labs. It helps the students understand how a stretched wire produces sound, and how exactly that sound can be precisely changed. A sonometer, derived from an ancient instrument called monochord, connects music to mathematics. In real life it is used to tune various musical instruments.

Objectives of the Sonometer Experiment

Study and understand the relationship between the vibrating frequency and the length of a given wire under constant tension using a monochord sonometer and thereby plot the f vs 1/l graph and study its characteristics.
Study and understand the relationship between the length of a given wire and the tension applied to it for a constant frequency using a monochord sonometer and thereby plot the l2 vs T graph and study its characteristics.

Explanation : Basically, in the sonometer experiment what we will do is this. First, we will set up the sonometer with the given wire and weights. Then we will try to match the vibrating tones of a tuning fork and the wire, at which point they will resonate together. The best way to do this is to place a piece of paper on the taut wire and see at which length the wire vibrates so much that it throws off the paper.

We would record those data about the length, tension, and frequency ‒ and then we’d proceed with another setting of the same. At the end of the experiment, we will plot all of those data on graph papers so that we can figure out how they are related to each other.

Let’s move forward to the sonometer experiment procedure.

Procedure of the Sonometer Experiment

A. equipment and material.

A sonometer apparatus of good quality ( you can get one here )
A given steel of nichrome wire ‒ about 1.5 meter of it. They are generally quite thin, like 0.5 mm. Guitar wires, obviously, produce the best results.
A weight hanger suitable to hold up to 2 kgs
A set of weights from 100gm to 1 kg
A set of tuning forks with known frequencies
A rubber pad to hit the tuning fork with
Some pieces of paper

B. Setting up the experiment

A sonometer is basically a long wooden box with a system on top to mount a wire tightly. You will need to set this up at the beginning of the sonometer experiment.

Place the sonometer box on top of your dry lab workbench in your physics practical class. Place it in a way so that the end with the pulley is flush with the open side of the table, so that something can hang from the pulley.
Get the wire you have to experiment with and verify that you have enough length of it. You’ll need about one and a half meters of it. The teacher will already know the mass per unit length of the material of the wire, but you may have to figure it out yourself using a scale and a precision balance .
Attach one end of the wire to the hook at the end of the sonometer apparatus. Make sure it is strongly attached.
Attach the weight hanger to the other end of the wire.
Sling the wire over the pulley on the other end of the sonometer. Place a little weight on the weight hanger so that the wire is stretched tightly over the wood box.
Place the bridges on the sonometer, under the wire, so that they hold up the wire between them and the wire is stretched tightly between them. Pull them apart to the positions farthest from each other (at the end of the scale).

That’s all there is to it ‒ your sonometer is all set up now for work.

C. Step-by-step experiment process

Now that you have your sonometer all set up and ready, you can begin the sonometer experiment. Follow these steps to do it properly.

Take your notebook and prepare two tables like those given below. You can note all your experiment data here.
Note: if you are working with a steel wire, we recommend starting with 500 grams. You can increase it to nearly two kilos. But this range would be different for other kinds of wires. Please ask your lab instructor to advise you on the tensile strength of the wire.
Take a small rectangular piece of paper (about the length of your finger) and fold it to make it an inverted V. Mount it in the middle of the wire so it hangs there loosely.
Take a tuning fork and note down its frequency.
Hit the tuning fork with the rubber pad to set it vibrating.
Touch the handle of the vibrating tuning fork to the sonometer box. The wire will start vibrating in response to the vibrations coming from the tuning fork.
Chances are, that is not what you’ll get at first. It will take a lot of tries to get the perfect resonance. The nature of the wire also changes the difficulty of the experiment.

In reality, it’s nigh impossible to get a so-called sweet spot. All you’ll get is a region. So, you will have to take two length readings, l1 and l2 . Take one reading while you are decreasing the gap between the bridges and the wire starts vibrating, and take the other reading while you are increasing the gap and the wire stops vibrating. Then the calculation length l would be the mean of these two measures.
Note down the positions of the bridges from the scale beside them. Also note the weight.
Vary the weight and perform the same process again and again, to get some satisfactory results with the same tuning fork.
Now keep a good amount of weight fixed in the hanger (we recommend 1 kg for a steel wire). With that unchanged, use up all the other tuning forks from the set and do the same process.

Sonometer Experiment Data

To perform this physics class 11 experiment, you will need to record the data in a specific format. Please make the following tables in your notebook and use the data from the experiment above to fill it up.

Sl. no.	Tuning fork frequency,	Resonating wire length			(in cm-1)
Sl. no.	Tuning fork frequency,	Increasing length,	Decreasing Length	Mean length =(l1+l2)/2	(in cm-1)
1.
2.
3.
4.

Sl. no.	Load, in Kg	Tension, = in N	Resonating wire length			in cm2	(in cm2/N)
Sl. no.	Load, in Kg	Tension, = in N	Increasing length,	Decreasing Length	Mean length =(l1+l2)/2	in cm2	(in cm2/N)
1.
2.
3.
4.

You can take as many data rows as you need, not only 4. Now plot the data in the graphs ( f vs 1/l ) and ( l2 vs T ). Both graphs should come out to be almost near straight lines.

We have taken some measurements in our system and the results are as follows:

From the above graphs, you can see that the results come more or less in a straight line. This shows clearly how the length of a vibrating wire, the tension applied to it, and its frequency depend upon each other.

All the equipment you need to perform the sonometer experiment can be found in our CBSE/ICSE/State board lab equipment packages . The physics lab equipment package we have has the sonometer apparatus (teakwood), various kinds of wires, weights, and tuning forks.

It has been a pleasure describing the sonometer theory and practical to you! Please leave your thoughts and suggestions in the comment section given below. Happy learning!

Relationship between frequency and length of wire under constant tension using Sonometer

Introduction.

In the experiment for studying the relationship that exists between length and the tension with respect to the frequency of the sound wave, the device of sonometer is important. It is a device that follows the Law of tension, where it is directly proportional to the frequency associated with the given string. This device is a hollow rectangular box made out of wood and has a length of more than 1 meter. To the end of the sonometer, there lies a hook and on the other side, it has a pulley. This experiment is conducted to verify the Law of tension with the help of the device sonometer.

Aim of the experiment and materials required

The aim of this experiment is to show whether the Law of tension is truly exhibited by the sonometer or not and the relationship that lies between tension and length for the given wire having constant frequencies. However, in conducting this experiment, several materials are required that include the followings, device of sonometer, screw gauge, rubber pad, paper rider, slotted weights of 7 and 1/2 kg, meter scale, hanger 1/2 kg and lastly, a set for 8 tuning forks (Chauhan et al. 2021). This experiment will not only prove the Law of tension but will also help in learning the use of the device sonometer.

Theory and formula

The theory that is applied in this experimentation shows a few conditions, suppose the given string is plucked from its centre, which is stretched and fixed by two pints at each end, then there occurs vibration. This vibration seems to move out in the exact opposite direction along with the given string (Chauhan et al. 2021). This occurs due to the generation and travelling of the transverse wave through the string. For example, if the length of the string is l, and it is stretched with a tension (T) and has a mass (m) per unit length, then the frequency of the vibration is expressed as, $\mathrm{F\: = \:1/2l\sqrt{T/m}}$ . In this expression, F is considered as the constant, and in turn, it states that, m and $\mathrm{\sqrt{T/1}}$ is also constant.

Figure 1: Study of the Relation between the Length of Given Wire and Tension for Constant Frequency Using Sonometer

In conducting this experiment, certain steps are to be followed as follows. The first step states the setting of the sonometer on the tabletop, giving it a weight of 4 kg. The hanger used in this experiment needs to have a maximum weight that is suitable for carrying out the experiment. However, for this experiment, a frictionless pulley is used. Need to move the wooden bridges in order to maximize the length of the given wire. Selections of 256 Hz for forks are set, that is to strike against the rubber pad leading to generating vibrations. Wire AB is vibrated by plucking. Following these procedures, record the changes that are noticed at the time of observation.

Observations, Calculations and Model graph

Observations during this experiment are to be recorded in a tabular format as given below. Need to consider the frequency of the tuning fork F, at constant, which is 256 Hz. Record the tension and the change in length for the given experiment.

Figure 2: Relation between Tension T (N) and Length $\mathrm{l^2 \:(cm^2)}$

In order to calculate, certain things to be considered, one needs to find the length (L) and record the mean length in the column. Identify the $\mathrm{l^2}$ and $\mathrm{l^2/T}$. In the next step, I need to plot T and l2 along the x-axis and y-axis.

Reading no.	Mass (M) in kg	Tension (T)= Mg in N	$\mathrm{\sqrt{T}}$	Vibrating length (l) in m	$\mathrm{\sqrt{T/1}}$

Figure 3: Observation table Variation of resonant length with tension

Precautionary measures and possible sources of error

Certain precautionary measures are to be used, the first is to make use of frictionless pulleys, and the wire needs to be of uniform cross-section. Usage of rubber pad is necessary, for calculations, length can be increased and decreased (Amrita.olabs.edu, 2022). Rigid wire and sharp wooden bridges should not be used. Correct weights should not be used.

In this tutorial, a detailed discussion has been conducted on the experiment that defines the Law of tension. However, this experiment makes use of a sonometer, which is quite important for recoding pf the vibrations that are generated by plucking the middle part of the given string. However, specific procedures are to be maintained with certain precautionary measures, as, if errors lie during the experiment, then readings might not be appropriate.

Q1. What are a sonometer and its usage in daily life?

Ans. Sonometer is a device that is used for defining the intricate relationship that exists between the frequencies for the sound produced by the string at the time of plucking in the experiment. The device sonometer is used for diagnostic purposes, such as to measure density, tension and frequency for vibrations. This device is used in identifying the density of bone and hearing capacity.

Q2. What is the law of tension?

Ans. Law of tension is defined as the prime frequency of vibrations of the respective string that is directly proportional to the square root of tension, when the length of vibration and mass per unit length is considered to be a constant. It is expressed as $\mathrm{v\:\varpropto\:1/l}$, where, l is the length.

Q3. What is defined as the pitch of the sound and what factors affect its pitch?

Ans. Pitch of sound can be defined as the sound sensations that are perceived by the human ear, depending upon the aspects of frequency of vibrations. The factors that are responsible for affecting the pitch of sound includes, tension (T), mass per unit length and lastly, length (L).

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A Sonometer is a device which uses the concept of standing waves to measure the mass per unit length or density of a wire. A sonometer is also called a Monochord .

The first reference to a sonometer or monochord was found in the writings of the Sumerian empire . Some believe that Pythagoras reinvented the sonometer in the sixth century BCE. Robert Fludd invented the “Celestial Monochord” in 1618, which could be used as a musical tuner. The motivation for the development of the sonometer since then was to create a musical tuning device.

Sonometer Diagram

A Schematic diagram of a Sonometer setup

Sonometer Construction

As shown in above figure, a monochord mainly consists of three parts:

1. The Wire, Weights and Pulley Apparatus

A wire is stretched along the soundbox, between the hinge A and the pulley D . The wire turns around the pulley to let a weight W hang on the end of the wire.

2. The Soundbox

The soundbox is a cuboidal box made of wood, upon which other parts of the sonometer are built. The function of the sonometer box is to amplify the sound of the tuning fork.

The sonometer box also has a graduated ruler along its length to measure distances on the box.

3. The Knife Edges

The knife edges are made to let the user change the length of the wire responsible for producing standing waves. So, changing the positions of the knife edges changes the wavelength of the standing waves. In the figure, knife edges are denoted by C and B triangles.

Sonometer Working: The Theory of Standing Waves

We can assume that there are two parts of the sonometer wire :

(a) The part of the wire between the knife edges and, (b) The part of the wire other than the part between the knife edges.

Only part (a) is involved in a discussion of standing waves. Part (b) is the other supporting part, attached to the hinge A and weights W.

The waves produced on a stretched string are transverse. Standing waves can be considered as waves with wavelengths such that their antinodes coincide with the knife edges’ placement.

Mathematically this could be represented as:

Where, $l$ is the length of the wire between the knife edges and $\lambda$ is the wavelength of the transverse wave.

For $n = 1$,

$l = \frac{ \lambda }{ 2 } \Rightarrow \lambda = 2 l$ is called the fundamental mode of vibration .

$v = \nu \lambda$

$v = \sqrt{ \frac{T}{m} }$

$v \rightarrow$ Velocity of the transverse wave $\nu \rightarrow$ Frequency of the transverse wave $T \rightarrow$ Tension in the wire $m \rightarrow$ Mass per unit length of the wire

We get for the fundamental mode,

$\sqrt{ \frac{T}{m} } = \nu \times 2 l$

$\nu = \frac{1}{2l} \sqrt{ \frac{T}{m} }$

Sonometer Experiment

Aim: To find the frequency of the AC mains using a sonometer.

E quipment Required: A sonometer apparatus, weights, step-down transformer, horseshoe magnet, connecting wires and AC mains power supply.

Circuit or Schematic Diagram:

A Schematic and Circuit Diagram of AC Sonometer Setup

Procedure:

As shown in the circuit diagram, connect the primary coil terminals of the transformer to AC mains. Similarly, connect the secondary coil terminals to the end of the sonometer wire(one terminal at the hook/hinge; the other at the pulley).
Place the horseshoe magnet symmetrically in the middle of the two knife edges. Place the magnet T so that the magnet’s magnetic field is perpendicular to the length of the wire.
Hang a mass M at the end of the wire, and keeping it fixed, adjust the position of the knife edges so that the sound produced from the sonometer box is maximum. This could be confirmed when the paper rider R on the wire falls off. It represents maximum amplitude.
Measure this length and repeat step 3 for different masses.
Using a screw gauge, find the diameter and hence the wire’s cross-section area. Using the information about the density of the wire, find the mass per unit length m of the wire.

$A = \frac{\pi D^2}{4}$

$m = \rho A$

Where, $A \rightarrow$ Area of cross section $D \rightarrow$ Diameter $\rho \rightarrow$ Density of the wire

Observations Table:

Precautions:

The mass M also contains the mass of the hangers. So make sure to take that into account.
The wire should be uniform and free of any kinks or bends.
When measuring the diameter of the wire with the help of a screw gauge, make corrections for the zero error. Take at least three readings and use the average of these readings.
Make sure that the thickness of the sonometer wire is appropriate for the current passing through it. Use a rheostat if necessary.

We require a magnet which provides a uniform magnetic perpendicular to the length of the wire. A horseshoe magnet is a perfect choice because the magnetic field is constant between the magnet’s arms.

A brass wire is used in the sonometer experiment. A brass wire is non-magnetic. Therefore, it is a natural choice.

Standing waves of Transverse nature are produced in the sonometer wire.

The sonometer box is built to have a broad resonance peak around a few tens of Hertz. So vibrations around a few tens of Hertz are near the resonance peak. The energy transfer is good in this range, and we hear a loud sound due to the resonating air column inside the sonometer box.

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Prayag Yadav

Science Practicals 11 & 12

Search this blog, class 12 physics practical reading to find the frequency of the a.c. mains with a sonometer., apparatus required.

v= (1/2l) ×√(T/m)

Setup for finding frequency of A.C. mains by sonometer

Observation & Calculations

Precautions

Sources of errors, post a comment.

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Sonometer Experiment

Post author By Hemant More
Post date February 4, 2020
2 Comments on Sonometer Experiment

Science > Physics > Stationary Waves > Sonometer Experiment

In this article, we shall study construction and working of sonometer, and its use to verify the laws of string.

Laws of Vibrating String:

Law of Length:

If the tension in the string and its mass per unit length of wire remains constant, then the frequency of transverse vibration of a stretched string is inversely proportional to the vibrating length.

Law of Tension:

Law of Mass:

Construction:

A sonometer consists of a hollow rectangular wooden box to which a uniform wire is attached at one end. The other end of the wire is passed over two horizontal knife edges or bridges and then over a pulley. A weight hanger is suspended from the free end of the wire. By placing different weights in the weight hanger, the tension in the wire can be suitably adjusted.

The points at which the wire rests on the knife edges cannot vibrate at all. Hence, when the wire is set up into vibrations, these two points become nodes and the wire vibrates in the fundamental mode. The frequency of vibration of the wire can be varied by either changing the positions of the knife edges by changing the vibrating length or by placing different weights in the pan by changing the tension.

Use of Sonometer to Determine the Unknown Frequency of a Tuning Fork:

To determine the unknown frequency of a tuning fork, the tension T in the wire is kept constant and the vibrating length between the knife edges is so adjusted, that the fundamental frequency of the wire becomes the same as that of the fork. To test this a small paper rider is placed on the wire midway between the knife edges where an antinode is formed.

The fork is set up into vibration and its stem is placed on the wooden box. The length of the wire is adjusted till it vibrates in unison with the fork. When this happens, the centre of the vibrating wire vibrates with maximum amplitude due to resonance, and the paper rider is thrown off. Then the frequency of the tuning fork which is the same as the fundamental frequency of the wire is given by

Where ‘m’ is the mass per unit length of the wire. The frequency of the fork is determined, knowing l , T, and m.

Use of Sonometer to Verify the Law of Length:

To verify , the given wire (m = constant) is kept under constant tension (T = constant). A set of tuning forks having different frequencies n 1 , n 2 , n 3 , n 4, etc. is taken. The length of the wire, vibrating in unison with each fork, is determined in turn using a paper rider or hearing beats. Let the lengths corresponding to the frequencies be l1, l2, l3, l4, etc.

Then, it is found that, within the limits of experimental error, n 1 l 1 = n 2 l 2 = n 3 l 3 = n 4 l 4 = constant. Thus in general n l = constant or n ∝ 1/ l. If a graph of n against 1/ l is plotted, it comes out as a straight line.

Use of Sonometer to Verify the Law of Tension:

If the vibrating length and mass per unit length of wire remain constant then, the frequency of transverse vibration of a stretched string is directly proportional to the square root of the tension in the string. To verify the law, the vibrating length of the given wire and linear density ‘m’ is constant. A set of tuning forks having different frequencies n 1 , n 2 , n 3 , n 4 etc. is taken.

By adjusting the tension T, each fork is made to vibrate in unison with the fixed length of the wire, one after the other. Let the tensions corresponding to the frequencies n1, n2, n3, n4, etc. be T1, T2, T3, T4, etc. respectively. Then it is found that, within limits of experimental error.

A graph of n² against T comes out as a straight line.

Use of Sonometer to Verify the Law of Mass:

If the vibrating length and tension in the string remain constant then, the frequency of transverse vibration of a stretched string is inversely proportional to the square root of its mass per unit length. This law cannot be verified directly, as either ‘n’ nor ‘m’ can be varied continuously as in the case of l or T. Therefore, this law is verified indirectly as follows. The relation can be written as

Then, to verify the law, we must show that when n and T are kept constant. A number of wires having linear densities m 1 , m 2 , m 3 , m 4, etc. are taken. Each one of them is subjected to the same tension T. Then, using a given tuning fork (n = constant) each wire is made to vibrate in unison with the fork, by adjusting its length. Let l1, l2, l3, l4, etc. be the vibrating length corresponding to linear densities m1, m2, m3, m4, etc. respectively.

Then it is found that, within the limits of experimental error.

Hence the law is indirectly verified.

Previous Topic: Vibrations of String

Next Topic: Melde’s Experiment

Tags all hamonics , Antinode , Displacement antinode , Displacement node , end correction , Experiment , First overtone , Fundamental frequency , Fundamental mode , harmonic , Law of isochromism , Law of length , Law of mass , Law of tension , Mechanical wave , Node , odd harmonics , overtone , pipe open at both end , pipe open at one end , Pressure antinode , Pressure node , Progressive wave , Reflection of wave , Second harmonic , Second overtone , Sonometer , Stationary wave , Stationary waves , Third harmonic , Vibrating string , Vibration of air column , Wave

2 replies on “Sonometer Experiment”

Excellent explaination it is.

Thank you so much ♥️

To Study the Relation Between the Length of a Given Wire and Tension for Constant Frequency Using Sonometer

November 28, 2016 by Bhagya

Physics Lab Manual NCERT Solutions Class 11 Physics Sample Papers

Aim To study the relation between the length of a given wire and tension for constant frequency using sonometer.

Apparatus A sonometer, a set of eight tuning forks, 1\2 kg hanger, seven 1\2 kg slotted weights, rubber pad, paper rider, metre scale, screw gauge.

to-study-the-relation-between-the-length-of-a-given-wire-and-tension-for-constant-frequency-using-sonometer-1

Procedure 1. Repeat steps 1 to 13 of previous part with a tuning fork of frequency 256 Hz and load 4 kg. 2. Remove slotted weights one by one to reduce load to 3.5 kg, 3 kg,……, 1 kg and repeat above steps with same tuning fork. 3. Record your observations as given below.

to-study-the-relation-between-the-length-of-a-given-wire-and-tension-for-constant-frequency-using-sonometer-4

Precautions

Pulley should be friction less.
Wires used should be kink less and of uniform cross-section.
Loading of wires should not be beyond elastic limit keep max. load = one-third of breaking load.
Tuning fork should be vibrated by striking its prongs against soft rubber pad.
Readings for length decreasing and increasing should be noted and their mean used in calculations.
To transfer vibrations of the tuning fork to the wire through sonometer board, lower end of handle of tuning fork should be touched gently with sonometer board.
Weight of hanger should be included in the load.
Load should be removed after the experiment.

Sources of error

Wire may not be rigid and of uniform cross-sectional area.
Pulley may not be friction less.
Weights used may not be correct.
Bridges may not be sharp.

Question. 1. Define superposition of waves. Answer. The phenomena of intermixing of two or more waves to produce a new wave, is called superposition of waves.

Question. 2. State superposition principle. Answer. It states “the resultant displacement of a particle is equal to the vector sum of the individual displacements given to it by the superposing waves.

Question. 3. Define stationary waves. Answer. Superposition of two waves of same frequency and same amplitude and travelling with same velocity in opposite direction produces stationary waves.

Question. 4. Under what situation are stationary waves produced ? Answer. Stationary waves are produced when a reflected wave gets superposed with its incident wave.

Question. 5. Define nodes. Answer. In stationary wave, the regions of zero displacement and maximum strain, are called nodes. They are denoted by N.

Question. 6. What is the distance between two consecutive nodes ? Answer. The distance between the two consecutive nodes is equal to half the wavelength of the stationary waves. NM=λ/2

Question. 7. Define antinode. Answer. In a stationary wave, the regions of maximum displacement and zero strain, are called antinodes. They are denoted by A.

Question. 8. What is the distance between two consecutive antinodes ? Answer. The distance between two consecutive antinode is equal to half the wavelength of the stationary waves. AA=λ/2

Question. 9. What is the distance between a node and nearest antinode ? Answer. The distance between a node and nearest antinodes is equal to quarter wavelength of the stationary waves. NA=λ/4

Question. 10. How can transverse stationary waves be produced in a stretched string ? Answer. Transverse stationary waves can be produced in a string by stretching it with a tension and fixing it at its two ends. When plucked in the middle, transverse stationary waves are produced in it.

Question. 11. What is fundamental frequency of a vibrating body ? Answer. The least frequency of a vibrating body, is called its fundamental frequency. It is also called first harmonic.

Question. 12. What does the number of a harmonic represent ? Answer. The number of a harmonic gives the ratio of its frequency with fundamental frequency.

Question. 13. What is an overtone ? Answer. All frequencies of a vibrating body, other than the fundamental, are called overtones.

Question. 14. How are the overtones numbered ? Answer. They are numbered in order of their increasing frequency after the fundamental.

Question. 21. How can longitudinal stationary waves be produced in a rod ? Answer. Longitudinal stationary waves can be produced in a rod by clamping it and then rubbing it along its length by a resined cloth.

Question. 22. Describe different cases of vibration of a clamped rod. Answer. Different cases are : 1. Rod clamped at one end, —> l = λ/4. 2. Rod clamped in the middle, —> l=λ/2. 3. Rod clamped both ends, —> l = λ/2. 4. Rod clamped at a point at quarter length distance from one end—> l = λ.

Question. 23. Define pitch of a sound. Answer. It is a sensation which makes us feel whether the sound is shrill or grave (hoarse).

Question. 24. On what factor pitch depends ? Answer. Pitch depends upon frequency of sound.

Question. 25. Which sound is shrill ? Answer. A sound of more frequency has a high pitch. The sound is shrill.

Question. 26. Give some example of shrill sound. Answer. Sound produced by the humming of a mosquito is shrill.

Question. 27. Which sound is grave (or hoarse) ? Answer. A sound of less frequency has a low pitch. The sound is grave (or hoarse).

Question. 28. Give some example of grave (hoarse) sound. . Answer. The sound produced by the roaring of a lion is grave (or hoarse).

Question. 29. What is a tuning fork? Answer. It is a source of standard frequency useful for sound experiment with sonometer or resonance tube.

Question. 30. How do its different parts vibrate ? Answer. Its prongs vibrates in transverse mode, while its handle vibrates in longitudinal mode.

Question. 31. How is it that its prongs stop vibrating simply on touching with a finger, while its handle continues vibrating even when held in hand tightly ? Answer. It is because, vibrations of prongs are transverse and those of handle are longitudinal.

Question. 32. What is a sonometer ? Answer. It is a hollow and rectangular wooden box.

Question. 33. Why are there two holes in the side of the sonometer ? Answer. To make air vibrating inside the box to come in contact of outside air to enhance vibrations.

Question. 34. How sonometer board helps in experiment ? Answer. It increases loudness of sound produced by the vibrating wire.

Question. 35. What is the nature of vibrations of the sonometer board ? Answer. The vibrations of sonometer board are forced vibrations.

Question. 36. What is the nature of vibrations of the tuning fork ? Answer. The vibrations of the tuning fork are free vibrations.

Question. 37. What is the nature of vibrations of the sonometer wire when the rider falls ? Answer. The vibrations are resonant vibrations.

Question. 38. What is meant by resonant vibrations of the wire ? Answer. It means that frequency of the wire equals tuning fork frequency.

to-study-the-relation-between-the-length-of-a-given-wire-and-tension-for-constant-frequency-using-sonometer-13

End correction Rayleigh found that reflection of sound at open end of the tube does not take place from the position of the edge of the tube but from a slightly higher region. It is because the tube medium (air) continues even beyond the edge. According to Rayleigh, the distance of the region of reflection from the edge of the tube is 0.3 D, where D is the internal diameter of the tube. This distance is called end correction and is represented by the symbol x, i.e., x = 0.3 D. Applying end correction, we get l + x = λ/4 λ = 4(l + x) Hence, formula for velocity of sound becomes, υ = 4 v(l +x).

to-study-the-relation-between-the-length-of-a-given-wire-and-tension-for-constant-frequency-using-sonometer-16

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Sonometer – Law of Tension & Law of Length

Physics Practicals Class 11

Sonometer – Law of Tension & Law of Length

Teach science experiments in a gamified way
Boost conceptual clarity and knowledge retention
Aligned with National Education Policy 2020
Helpful in getting NAAC accreditation
CBSE, ICSE, and state boards aligned curricula
Engaging simulations with easy-to-teach instructions

About Simulation

During the simulation, you will gain a comprehensive understanding of resonance and resonant frequency, exploring the laws of length and tension using the sonometer.
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Published: 25 September 2024

The genetic architecture of protein stability

Andre J. Faure ORCID: orcid.org/0000-0002-4471-5994 1 nAff5 ,
Aina Martí-Aranda 1 , 2 ,
Cristina Hidalgo-Carcedo ORCID: orcid.org/0000-0003-3571-2449 1 ,
Antoni Beltran 1 ,
Jörn M. Schmiedel ORCID: orcid.org/0000-0002-6842-9837 1 nAff6 &
Ben Lehner ORCID: orcid.org/0000-0002-8817-1124 1 , 2 , 3 , 4

Nature ( 2024 ) Cite this article

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Computational biology and bioinformatics

There are more ways to synthesize a 100-amino acid (aa) protein (20 100 ) than there are atoms in the universe. Only a very small fraction of such a vast sequence space can ever be experimentally or computationally surveyed. Deep neural networks are increasingly being used to navigate high-dimensional sequence spaces 1 . However, these models are extremely complicated. Here, by experimentally sampling from sequence spaces larger than 10 10 , we show that the genetic architecture of at least some proteins is remarkably simple, allowing accurate genetic prediction in high-dimensional sequence spaces with fully interpretable energy models. These models capture the nonlinear relationships between free energies and phenotypes but otherwise consist of additive free energy changes with a small contribution from pairwise energetic couplings. These energetic couplings are sparse and associated with structural contacts and backbone proximity. Our results indicate that protein genetics is actually both rather simple and intelligible.

Massively parallel experiments allow the effects of single aa changes in proteins to be comprehensively quantified 2 , 3 . Similarly, experimental analysis of double mutants is feasible, at least for small proteins 4 , 5 . The analysis of higher-order mutants, however, quickly becomes infeasible owing to the combinatorial explosion of possible genotypes. For example, the number of ways to combine one mutation at 34 different sites in a protein is 2 34 ≈ 1.7 × 10 10 . Experimental exploration of such a large number of genotypes is extremely challenging 6 given current technology, which—so far—has experimentally analysed sequence spaces up to about 10 6 (refs. 4 , 7 ).

Moreover, combining random mutations in even moderate numbers nearly always results in non-functional proteins 8 , 9 . For example, only 2–8% of 5 aa variants and <0.2% of 10 aa variants in a small protein domain are expected to be folded if energies combine additively ( n = 2 domains; Fig. 1a and Extended Data Fig. 1a ). Sampling even tens of millions of random combinatorial genotypes in most proteins will therefore provide almost no information about genetic architecture—the set of rules that govern how mutations combine to determine phenotypes—and will not be useful for training and evaluating predictive models beyond testing the trivial prediction that most genotypes are unfolded.

a , Violin plot showing distributions of simulated AbundancePCA growth rates (assuming additivity of individual inferred folding free energy changes 23 ) versus number of random aa substitutions ( n = 100,000). Violins are scaled to have the same maximum width. b , DMS data, energy model and algorithm used to select a set of single aa substitutions for combinatorial mutagenesis. A shallow double-mutant library of GRB2-SH3 protein variants was assayed by AbundancePCA (see panel c ) and BindingPCA (see Fig. 4b ; in combination referred to as ddPCA), followed by energy modelling to infer single aa substitution free energy changes of folding and binding 23 . We used this model together with a greedy algorithm to select a set of 34 single aa substitutions that, when combined, would simultaneously maximize both the predicted AbundancePCA and BindingPCA growth rates, that is, preserving both fold and function. 3D structure of GRB2-SH3 (PDB: 2VWF) indicating the 34 combinatorially mutated residues (orange) and GAB2 ligand (blue) is shown on the right. c , Overview of AbundancePCA on the protein of interest (GRB2-SH3) 23 . yes, yeast growth; no, yeast growth defect; DHF, dihydrofolate; THF, tetrahydrofolate. d , Scatter plots showing the reproducibility of fitness estimates from triplicate AbundancePCA experiments. Pearson’s r is indicated in red. Rep., biological replicate. e , Histogram showing the number of observed aa variants at increasing Hamming distances from the wild type (denoted by WT), for which the x axis is shared with panel f . f , Violin plot showing distributions of AbundancePCA growth rates inferred from deep sequencing data versus number of aa substitutions. In panels a and f , the percentage of folded protein variants (predicted fraction folded molecules > 0.5) is shown at each Hamming distance from the wild type.

One strategy for exploring high-dimensional sequence spaces is to use deep learning. Deep neural networks with millions of fitted parameters have proved successful for diverse prediction and protein design tasks, including predicting the effects of combinatorial mutants 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 . However, these models have extremely complicated and difficult to interpret architectures.

It could be that protein genotype–phenotype landscapes are complex, with many interactions between mutations required for accurate prediction. Alternatively, these landscapes might be much simpler, as suggested by energy measurements 22 and inferences 23 , 24 and the use of statistical models 25 , 26 . For us, a simple model is one with few parameters (so providing a large data compression) and parameters that are interpretable (so providing understanding).

Here we use an experimental design that enriches functional protein sequences to explore the genetic architecture of high-dimensional protein sequence spaces with more than 30 dimensions and more than 10 10 genotypes. We find that protein architectures are remarkably simple, with additive energy models providing very good predictive performance. Quantifying the pairwise energetic couplings between mutations further increases predictive power, providing excellent performance in high-dimensional genotype spaces. These couplings are sparse and related to protein 3D structures. The genetic architecture of at least some proteins is therefore very simple, with additive energetics and a small contribution from sparse pairwise structural couplings.

Sampling a 10 10 sequence space

We previously showed that the energetic effects of thousands of individual mutations on the stability of a protein can be measured en masse using pooled variant synthesis, selection and sequencing experiments 23 , 24 . In these experiments, the effect of each mutation on the cellular abundance of a protein is measured in the wild-type protein and also in a small number of variants with different fold stabilities. For example, using a shallow double-mutant library, we could infer the changes in Gibbs free energy of folding (ΔΔ G f ) for nearly all mutations (1,056/1,064 = 99%) in the C-terminal SH3 domain of the adaptor protein GRB2 (ref. 23 ). Similar massively parallel measurements of single-mutant fold stabilities have now been made for other signalling domains, including the oncoprotein KRAS 24 , and, in vitro, for many, mostly prokaryotic, small domains 7 .

Combining random mutations in the GRB2-SH3 domain very quickly results in unfolded proteins, with about 98% and more than 99.9% of genotypes with five and ten mutations expected to be unfolded (based on additive energies; see Fig. 1a ). This rapid decay of stability as mutations are combined is consistent with experimental measurements of the activity of other proteins 8 , 9 . To experimentally explore folded genotypes in high-dimensional sequence spaces, we therefore used a heuristic technique to enrich for conserved fold and function in combinatorial variants. For each possible starting single aa substitution, we iteratively selected further substitutions—one per residue position—that simultaneously maximizes the resulting combinatorial mutant’s predicted abundance and binding to an interaction partner (see Methods ). For GRB2-SH3, the largest set of mutations predicted to preserve both molecular phenotypes consisted of 34 single aa substitutions: 25 in surface residues (relative solvent-accessible surface area (RSASA) ≥ 0.25), three in the protein core (RSASA < 0.25) and six mutations in the GAB2 ligand binding interface (ligand distance < 5 Å; Fig. 1b , right).

We synthesized a library (‘library 1’) containing all combinations of these 34 mutants and quantified the cellular abundance of a sample of the 2 34 ≈ 1.7 × 10 10 genotypes using a highly validated pooled selection, abundance protein fragment complementation assay (AbundancePCA 23 , 24 , 27 ). In total, we obtained triplicate abundance measurements for 129,320 variants, which is 0.0007% of the sequence space. The measurements were highly reproducible (Pearson’s r > 0.91; Fig. 1d and see Methods ).

The symmetrical pod-like shape of the genotype frequency landscape, with the number of genotypes peaking at the intermediate Hamming distance of 17—that is, equidistant from the wild type (zeroth-order) and 34th-order mutant—is recapitulated in the experimentally sampled library (Fig. 1e ). Median abundance measurements decrease with increasing number of aa substitutions, but there are still thousands of genotypes with many mutations that, nevertheless, maintain abundance scores that are indistinguishable from that of the wild-type protein ( n = 2,706 with more than 20 mutations, two-sided z -test nominal P > 0.05; Fig. 1f ).

Genetic prediction with energy models

Quantifying the effects of a large number of multi-mutants allowed us to test the predictive performance of genotype–phenotype models in regions of the genetic landscape beyond the local neighbourhood used for training. For model building and evaluation, we restricted all analyses to variants with quantitative measurements in all three biological replicates ( n = 71,233). Notably, our original energy model (Fig. 2a ) trained on abundance and ligand binding selections (doubledeepPCA, ddPCA) quantifying the effects of single and double aa mutants only 23 explains as much as half of the fitness variance in combinatorial multi-mutants ( R 2 = 0.5; Fig. 2b , lower-right panel), for which most (94%) include at least 13 aa substitutions in the wild-type sequence. The only trained parameters in this simple model are Gibbs free energy terms for the wild type (Δ G f ) and single aa substitutions (ΔΔ G f ) and a two-parameter (affine) transformation relating the fraction of folded molecules to the AbundancePCA score (fitness; Fig. 2a ). That such a large proportion of phenotypic variance is explained by an additive energy model (no specific epistasis/genetic interactions) trained on genotypes containing only one or two genetic changes suggests that the energetic effects of mutations in proteins are largely context-independent.

a , Two-state equilibrium and corresponding neural network architecture used to fit thermodynamic model to AbundancePCA data (bottom, target and output data), thereby inferring the causal changes in free energy of folding associated with single aa substitutions (top, input values). Δ G f , Gibbs free energy of folding; K f , folding equilibrium constant; p f , fraction folded; g , nonlinear function of Δ G f ; R , gas constant; T , temperature in kelvin. b , Performance of first-order linear models (left column) and first-order energy models (right column) evaluated on GRB2-SH3 combinatorial AbundancePCA data. The top row indicates the results of models that were trained on a subset of the same combinatorial DMS data. The bottom row indicates the results of models that were trained on GRB2-SH3 ddPCA data consisting of single and double aa substitutions only 23 . R 2 is the proportion of variance explained. c , Nonlinear relationship (global epistasis) between observed AbundancePCA fitness and changes in free energy of folding. Thermodynamic model fit is shown in red. d , Comparisons of the model-inferred free energy changes to previously reported estimates using GRB2-SH3 ddPCA data 23 . Pearson’s r is shown. e , Performance of energy model that includes all first-order and second-order genetic interaction (energetic coupling) terms/coefficients. See Extended Data Fig. 2 for plots of the residuals versus fitted values for linear and energy models of the first and second order. f , Distributions of folding free energy changes (ΔΔ G , grey) and pairwise energetic couplings (ΔΔΔ G , red). WT, wild type.

On the other hand, a linear model—which implicitly assumes that mutation effects combine additively at the phenotypic level in multi-mutants—trained on the same ddPCA data performs much worse ( R 2 = 0.32). The linear model also systematically underestimates the observed phenotypic effects of mutant combinations (Fig. 2b , lower-left panel), a consequence of not accounting for the scaling of mutational effects owing to protein thermodynamics (global epistasis 28 , 29 , 30 ). For example, introducing a destabilizing mutation in an already-unfolded protein has no effect on the fraction of folded molecules (lower plateau of model in Fig. 2c ), which is not captured by a linear model. These results demonstrate a key advantage of fitting energy models: accounting for global epistasis improves the generalizability of predictions beyond the local neighbourhood of the training data.

Fitting linear and energy models to the combinatorial data improves the variance explained by 30% and 13%, respectively (Fig. 2b , upper panels), probably because of the greater amount of training data and (relevant) genetic backgrounds in which the effects of each single aa are quantified: about 50% of all variants in the library ( n ≈ 30,000) include—and therefore report on the effects of—any given one of the 34 single aa substitutions, that is, almost three orders of magnitude more measurements per single mutant compared with the relatively shallow (ddPCA) library. Although the fraction of variance explained by first-order linear and energy models is comparable ( R 2 = 0.62 and 0.63; Fig. 2b , upper panels), the biased regression residuals in the case of the linear model show that this model is less appropriate (Extended Data Fig. 2a ). The energy model provides an excellent fit to the data, faithfully capturing the global nonlinear relationship (global epistasis) between observed AbundancePCA fitness and inferred changes in free energy of folding (Δ G f ) (Fig. 2c ). There is also very good agreement between free energy changes (model parameters) inferred from combinatorial and ddPCA datasets (Pearson’s r = 0.87), but the former tend to be more extreme, once again demonstrating the use of assaying the effects of mutations in greater numbers of genetic backgrounds, thereby allowing their energetic effects to be more accurately estimated (Fig. 2d ).

Couplings improve genetic prediction

We next tested whether quantifying non-additive energetic couplings between mutations improved predictive performance. In our combinatorial dataset, each pair of mutations is present in a median of 17,923 genotypes, allowing robust measurement of second-order genetic interaction terms (energetic couplings, ΔΔΔ G f (refs. 31 , 32 )). Including all second-order energetic couplings improves model performance by an extra 9% ( R 2 = 0.72), consistent with expectations that pairwise effects are an important source of specific epistasis in proteins 32 (Fig. 2e ). Whereas first-order terms are stronger in magnitude and biased towards destabilizing effects, second-order energetic couplings tend to have milder effects centred on zero (Fig. 2f ).

Physical contacts and backbone proximity

Measured at the phenotypic level, genetic interactions in proteins have previously been shown to reflect—at least in part—protein structures 4 , 33 , 34 , 35 , 36 . Combining combinatorial deep mutational scanning with thermodynamic modelling allowed us to infer a total of 561 pairwise energetic couplings, providing an opportunity to interrogate their mechanistic origins and relationship with protein structure. Comparing coupling energy magnitude (absolute folding ΔΔΔ G f ) with the 3D distance separating mutation pairs in the folded structure (minimal side-chain heavy-atom distance) reveals an L-shaped distribution with the strongest energetic couplings occurring between structurally proximal residues (Fig. 3a ; see also Extended Data Fig. 3a ). The top five energetic couplings all involve pairs of residues within 5.5 Å and 15 of the top 20 (75%) energetic couplings involve pairs separated by less than 8 Å. Although there is a weak anticorrelation between contact distance and coupling energy strength (Spearman’s ρ = −0.12; Fig. 3a ), this trend breaks down for pairs that are not proximal in the primary sequence (Spearman’s ρ = −0.02, backbone distance >5 residues).

a , Relationship between folding coupling energy strength and minimal inter-residue side-chain heavy-atom distance. The mean is shown and error bars indicate 95% confidence intervals from a Monte Carlo simulation approach ( n = 10 experiments). Points are coloured by binned inter-residue distances (see legend in panel b ). Spearman’s ρ is shown for all couplings (top value), as well as those involving pairs of residues separated by more than five residues in the primary sequence (bottom value). Core residues are indicated as triangles. b , Relationship between folding coupling energy strength and linear sequence (backbone) distance in number of residues. The measure of centre and error bars are as defined in panel a . c , Interaction matrix indicating folding energy coupling strength, as well as pairwise structural contacts in GRB2-SH3 (Protein Data Bank (PDB): 2VWF; minimal side-chain heavy-atom distance < 8 Å, black circles). Grey cells indicate missing values (non-mutated residues) and constitutive single aa substitutions are indicated in the x -axis and y -axis labels. Black diagonal lines demarcate residue pairs that are distal in the primary sequence (backbone distance > 5 residues). Reference secondary structure elements (arrow, beta strand) are shown at the top and right. d , 3D structure of GRB2-SH3 (PDB: 2VWF) indicating the top ten energetic couplings with black dashed lines. Combinatorially mutated residues are shown in orange. e , Bar plot showing ranked features from the linear model to predict folding coupling strength. Bar width indicates coefficient significance ( P value from uncorrected two-sided t -test). Blue, positive coefficient; red, negative coefficient; grey, non-significant ( P > 0.05). f , Correlation between linear model predicted and observed folding coupling strength for training (top) and test (bottom) data. Pearson’s r is shown. The error bands represent the 95% confidence intervals for the predicted values.

On the other hand, comparing coupling strength with separation distance between residues in the primary sequence (along the peptide backbone) reveals a marked inverse relationship that extends over quite large distances (Spearman’s ρ = −0.28) and is robust to the exclusion of direct physical contacts between residues (<5 Å, Spearman’s ρ = −0.27; Fig. 3b and see also Extended Data Fig. 3b ). The interaction matrix in Fig. 3c summarizes these observations: the strongest energetic couplings coincide with direct physical contacts (black circles; see also Fig. 3d ) and energetic coupling strength decays along the protein backbone (Fig. 3c , near-diagonal versus far off-diagonal cells). The matrix also highlights physical interactions between secondary structural elements as hotspots for strong energetic couplings.

To disentangle the relative importance of these different potential structural determinants of energetic coupling strength, we gathered a collection of quantitative features describing both the number and the type of chemical bonds or interactions existing between the atoms of pairs of residues, as well as their relative positions in the folded structure (Fig. 3e ). A linear regression model based on these 12 structural features is predictive of coupling strength (Fig. 3f ; see Methods ). Notably, the same model performs similarly well when evaluated on a held out, non-overlapping set of inferred energetic couplings derived from an independent combinatorial mutagenesis experiment (‘library 3’, which is described below; Pearson’s r = 0.46, R 2 = 0.21). This suggests that, despite its simplicity, the integrated model captures structural determinants of energetic coupling strength and that energetic couplings are caused by structural interactions.

Couplings decay along the peptide chain

To directly test the hypothesis that inter-residue backbone distance is associated with energetic coupling strength independently of 3D contact distance, we designed a combinatorial saturation mutagenesis library involving all possible mutations at four physically proximal surface residues in the same secondary structure element (‘library 2’; Extended Data Figs. 4a and 5a and see Methods ). We reasoned that energetic couplings owing to the propagation of perturbations along the protein backbone should also be apparent among solvent-facing residues. In total, we obtained abundance measurements for 138,157 variants (86% of the sequence landscape) and the measurements were highly reproducible (Pearson’s r > 0.89; Extended Data Fig. 5b and see Methods ).

The single-mutant effects at these four residues have a larger range than those of the combinatorial library that was designed to conserve fold and function (Extended Data Fig. 5c,d ). Therefore, when combined in double, triple and quadruple mutants—the most numerous class—the result is a larger fraction of unfolded variants (Extended Data Fig. 5c–e ). A two-state thermodynamic model that includes all first-order and second-order coefficients provides an excellent fit to the data ( R 2 = 0.93; Extended Data Fig. 5e–g ) and inferred folding free energy changes (first-order terms) are highly correlated (Pearson’s r = 0.94) with those obtained previously using an independent shallow double-mutant library (Extended Data Fig. 5h ). Although the four mutated residues are physically proximal in 3D space, with all except one pair (H26:T44) separated by less than 5 Å (3.8–8.4 Å; Extended Data Fig. 5i ), their relative positions in the primary peptide sequence cover a large range (2–18 residues; Extended Data Fig. 5j ). There is no relationship between contact distance and folding coupling strength for these contacting residues (Spearman’s ρ = −0.05; Extended Data Fig. 5i ), whereas the relationship for backbone distance is significant (Spearman’s ρ = −0.41; Extended Data Fig. 5j ). Indeed, backbone distance is very well correlated with coupling strength when averaging energy terms per residue pair (Spearman’s ρ = −0.94; Extended Data Fig. 5j ). The relative position of aa residues in the primary protein sequence is therefore associated with coupling strength independently of their proximity in 3D space.

Higher-order mutants fold and function

Our experiments identified a large number of GRB2-SH3 genotypes containing many mutations that have high cellular abundance (for example, 25,564 genotypes containing more than five mutations; Fig. 1f ). To further confirm that abundant multi-mutants are correctly folded and functional, we performed a third combinatorial mutagenesis experiment in which we also tested the ability of GRB2-SH3 variants to bind to a peptide ligand using a protein–protein interaction assay (BindingPCA 23 , 24 , 37 ; Fig. 4a ). Recognition of the peptide ligand can only occur if the protein adopts its native conformation 38 (Fig. 1b,c ). We designed a library (library 3) consisting of all combinations of 15 single aa substitutions occurring within a 22-aa residue window, avoiding mutations in our original library in binding interface residues (minimal side-chain heavy-atom distance to the ligand < 5 Å; Extended Data Fig. 4b and also see Methods ). The library contains 2 15 (=32,768) variants and shares six single aa substitutions with our original 2 34 library. In total, we obtained binding measurements for 25,967 variants and abundance measurements for 31,936 variants (79% and 97% of the sequence landscape, respectively). The measurements were highly reproducible (Pearson’s r > 0.85 and 0.94 for binding and abundance, respectively; Fig. 4c , Extended Data Fig. 6a–f and also see Methods ).

a , 3D structure of GRB2-SH3 (PDB: 2VWF) indicating 15 combinatorially mutated residues in library 3 (orange) and GAB2 ligand (blue). b , Overview of BindingPCA of GRB2-SH3 binding to GAB2 (ref. 23 ). no, yeast growth defect; DHF, dihydrofolate; THF, tetrahydrofolate. c , Scatter plots showing the reproducibility of fitness estimates from triplicate BindingPCA experiments. Pearson’s r indicated in red. Rep., biological replicate. d , 2D density plots comparing abundance and binding fitness of third-order (left), sixth-order (middle) and ninth-order mutants (right). See Extended Data Fig. 6g for similar plots at all mutant orders. e , Three-state equilibrium and corresponding thermodynamic model. Δ G f , Gibbs free energy of folding; Δ G b , Gibbs free energy of binding; K f , folding equilibrium constant; K b , binding equilibrium constant; c , ligand concentration; p f , fraction folded; p fb , fraction folded and bound; R , gas constant; T , temperature in kelvin. f , Neural network architecture used to fit thermodynamic model to ddPCA data (bottom, target and output data), thereby inferring the causal changes in free energy of folding and binding associated with single aa substitutions (first-order terms) and pairwise (second-order) interaction terms (top, input values). Variables as per panel e and: g f , nonlinear function of Δ G f ; g fb , nonlinear function of Δ G f and Δ G b . g , Nonlinear relationships between observed AbundancePCA fitness and changes in free energy of folding (top) or BindingPCA fitness and free energies of both binding and folding (bottom). Thermodynamic model fit is shown in red. h , Performance of models fit to ddPCA data. R 2 , proportion of explained variance. i , Comparisons of the model-inferred single aa substitution free energy changes to previously reported estimates using GRB2-SH3 ddPCA data 23 . Pearson’s r is shown.

Plotting the changes in binding against the changes in abundance for third-order, sixth-order and ninth-order variants shows that most mutations altering binding also alter the concentration of the isolated domain, consistent with previous results and the expectation that changes in protein stability are a main cause of mutational effects on binding 23 , 24 , 39 (Fig. 4d and Extended Data Fig. 6g ). Notably, however, most higher-order mutants that have high abundance scores also bind the GAB2 ligand, indicating that they are correctly folded (Fig. 4d and Extended Data Fig. 6g ). For example, 4% (204/4,805) of variants containing nine mutations have abundance indistinguishable from that of the wild-type protein (nominal P > 0.05) and 96% (177/184) of these also bind the ligand (predicted fraction bound molecules > 0.5). Most of the abundant higher-order GRB2-SH3 mutants are thus correctly folded.

Multi-phenotype genetic prediction

The large number of genetic backgrounds in which both single and double aa mutant effects were measured for these two related molecular phenotypes is a rich source of data for thermodynamic modelling. First, considering only the abundance phenotype, we observe that an additive two-state thermodynamic model—with unfolded and folded energetic states—outperforms a linear model when evaluated on held-out variants ( R 2 = 0.93 versus 0.87; Extended Data Fig. 7a,b ). To attain similar predictive performance as the first-order energy model requires inclusion of both second-order and third-order genetic interaction terms in the linear model (Extended Data Fig. 7c,d ), representing a massive increase in model complexity (715 versus 16 parameters; that is, greater than 40-fold more). This greater complexity of models that use many specific pairwise and higher-order genetic interaction terms to capture global nonlinearities in data (global epistasis) has been referred to as ‘phantom epistasis’ 40 .

Next, extending previous work 23 , we used a neural network implementation of a three-state equilibrium model 41 —with unfolded, folded and bound energetic states (Fig. 4e )—to simultaneously infer the underlying causal free energy changes of both folding and binding (ΔΔ G f and ΔΔ G b ), as well as folding and binding energetic couplings (ΔΔΔ G f and ΔΔΔ G b ) (Fig. 4f ). The model fits the data extremely well (Fig. 4g ), explaining virtually all of the fitness variance (Fig. 4h ), and the inferred folding and binding free energy changes (first-order terms) are well correlated (Pearsons’s r = 0.9 and 0.7) with those obtained previously using an independent shallow double-mutant library 23 (Fig. 4i , double mutants). This is the first time, to our knowledge, that a large number of folding (ΔΔΔ G f ) and binding (ΔΔΔ G b ) energetic coupling terms have been measured for any protein.

Allostery in ligand-proximal residues

We observe that mutational effects on folding energy tend to be larger than those on binding energy (Fig. 5a ), recapitulating previous results 23 , 24 . Energetic couplings show the same pattern, with folding coupling energies tending to be larger in magnitude than binding energetic couplings (area under the curve = 0.7, n = 210, P = 3.6 × 10 −7 , two-sided Mann–Whitney U test, |ΔΔΔ G f | mean = 0.087, s.d. = 0.084, |ΔΔΔ G b | mean = 0.038, s.d. = 0.035; Fig. 5b ). As none of the mutations in this library occur in the binding interface, any notable effects on binding affinity must be through an allosteric mechanism 23 , 24 . Plotting absolute free energy changes against the 3D distance to the ligand shows a negative correlation as previously reported 23 , 24 (Spearman’s ρ = −0.46), with mutations in second-shell residues and residues adjacent (in the sequence) to binding interface residues highly enriched for strong allosteric effects on binding affinity (Fig. 5a ). Consistent with previous observations 23 , 24 , mutations at distal glycine residues have among the strongest effects on binding affinity.

a , Relationship between the absolute change in free energy of folding (top) and binding (bottom) and minimal side-chain heavy-atom distance to the ligand. Residues are coloured by their position in the structure relative to the binding interface, triangles indicate beta strand residues and connection lines indicate the strength of energetic couplings between aa pairs (see legend). Spearman’s ρ is shown. b , Interaction matrix indicating folding (top) and binding (bottom) coupling terms, as well as pairwise structural contacts in GRB2-SH3 (PDB: 2VWF; minimal side-chain heavy-atom distance < 8 Å, black circles). Grey cells indicate missing values (non-mutated residues) and constitutive single aa substitutions are indicated in the x -axis and y -axis labels (see panel a for axis label text colour key). Mutations in beta strand residues are indicated and couplings between beta strand residues are boxed. The bar plots above and to the right of the binding interaction matrix indicate the total number of pairwise physical interactions (<8 Å) involving each residue, with green bars indicating the fraction of interacting partners classified as second-shell residues. The strongest binding energetic coupling (P11A:G18C) is indicated by an arc.

Whereas the mutations with the strongest folding coupling energies are near-diagonal (closely spaced in the primary sequence), particularly between pairs of residues in the beta strand, the strongest binding coupling in the dataset is an interaction between residues P11 and G18 (Fig. 5b ). These two residues are proximal in 3D space (<8 Å) and constitute one of only two long-range physical contacts between the mutated residues (backbone distance > 5 residues; Fig. 5b ), suggesting that allosteric energetic couplings are also driven by structural contacts.

SRC kinase combinatorial mutagenesis

Finally, to further test the generality of our conclusions, we used the same greedy approach to design a library containing 2 15 (=32,768) variants in an unrelated and larger protein, the human proto-oncogene tyrosine-protein kinase Src (SRC). We obtained triplicate abundance measurements for 31,557 variants and the measurements were highly reproducible ( r > 0.86; Fig. 6b–d ). As for our three GRB2-SH3 combinatorial libraries, a second-order energy model was highly predictive of abundance changes ( R 2 = 0.87; Fig. 6e,f ), with energetic couplings predicted by both 3D spatial proximity (Fig. 6g ) and backbone proximity (Fig. 6h ). The consistency of these results in an unrelated full-length protein further supports their generality.

a , 3D structure of SRC (PDB: 2SRC) indicating the 15 combinatorially mutated residues in library 4 (orange) and ATP (blue). b , Scatter plots showing the reproducibility of fitness estimates from triplicate AbundancePCA experiments. Pearson’s r indicated in red. Rep., biological replicate. c , Histogram showing the number of observed aa variants at increasing Hamming distances from the wild type, in which the x axis is shared with panel d . d , Violin plot showing distributions of abundance growth rates inferred from deep sequencing data versus number of aa substitutions. The percentage of bound protein variants (predicted fraction bound molecules > 0.5) is shown at each Hamming distance from the wild type. e , Nonlinear relationship (global epistasis) between observed abundance fitness and changes in free energy of folding. Thermodynamic model fit is shown in red. f , Performance of energy model that includes all first-order and second-order genetic interaction (energetic coupling) terms/coefficients. g , Relationship between folding coupling energy strength and minimal inter-residue side-chain heavy-atom distance. The mean is shown and error bars indicate 95% confidence intervals from a Monte Carlo simulation approach ( n = 10 experiments). Points are coloured by binned inter-residue distances (see legend in panel h ). Spearman’s ρ is shown for all couplings (top value), as well as those involving pairs of residues separated by more than five residues in the primary sequence (bottom value). Core residues are indicated as triangles. h , Relationship between folding coupling energy strength and linear sequence (backbone) distance in number of residues.

By experimentally quantifying protein fold stability in samples from sequence spaces greater than 10 10 in size, we have shown here that the fundamental genetic architecture of at least some proteins is remarkably simple. Thermodynamic models in which the energetic effects of mutations are summed provide very good prediction of fold stability when tens of mutations are combined. Quantifying the pairwise energetic couplings between mutations further increases predictive power, providing very good performance in high-dimensional genotype spaces. The large number of energetic couplings quantified here reveals important principles about their origins: couplings are strongest between structurally contacting residues and coupling strength also decays along the protein backbone.

The energy models used here are very sparse and represent very large data compressions: up to about 10 8 (2 34 /34) for the additive models and up to about 10 7 (2 34 /596) for the models with energetic couplings. Analyses of previously published combinatorial protein mutagenesis datasets 8 , 41 , 42 , 43 , mutagenesis of a protein interaction interface 44 , hydrophobic protein cores 45 , an intrinsically disordered region 46 , a tRNA 43 , 47 and an alternatively spliced exon 40 suggest that this simplicity of genotype–phenotype landscapes is widely observed and probably a general principle of macromolecules and their molecular interactions.

Energy models are grounded in our understanding of protein thermodynamics and their simplicity and interpretability contrasts with the complexity and lack of mechanistic insight provided by deep neural networks. Predictive energy models are likely to have many applications, including for clinical variant effect interpretation 48 , pathogen and pandemic forecasting 49 and protein engineering for biotechnology 1 . An important challenge moving forward is how to efficiently quantify the free energy changes and energetic couplings for all mutations in proteins of interest. Quantifying mutational effects across diverse genetic backgrounds and homologous sequences may be an efficient way to achieve this 50 .

Our data do not rule out the importance of higher-order genetic interactions for protein stability. Rather, they show that, when global nonlinearities owing to cooperative protein folding are properly accounted for and measurements are averaged across genetic backgrounds, first-order and pairwise energetic couplings provide sufficient information for many prediction tasks. An important question to address in future work will be the extent to which higher-order energetic interactions become important in even larger sequence spaces, including in the ‘twilight zone’ of structurally homologous proteins with very low sequence identity. Indeed, the superior performance of our models in 2 15 -sized compared with in 2 34 -sized sequence spaces hints that higher-order interactions become increasingly important as sequences diverge. Simple experimental designs should be able to definitively address this question for a diversity of protein folds.

Combinatorial mutagenesis library designs

Combinatorial library 1.

Library 1 was designed using a computationally efficient greedy strategy to search for the largest number of single aa substitutions that, when combined, preserve both fold and function even in the highest-order mutants (Fig. 1b ). The algorithm used previously published ddPCA data and thermodynamic modelling results for GRB2-SH3, including inferred single aa substitution free energy changes of folding and binding for this protein 23 . We showed previously that this model—which assumes that individual inferred folding and binding free energy changes (ΔΔ G f and ΔΔ G b ) combine additively in multi-mutants—accurately predicts the effects of double aa substitutions 23 . Therefore, this same additive model was used to make predictions about the energetic and phenotypic effects of higher-order mutants explored in the greedy search.

First, the set of candidate single aa mutations was restricted to those with confident free energy changes, defined as those with 95% confidence intervals < 1 kcal mol −1 and whose effects were measured in at least 20 genetic backgrounds (that is, double aa mutations). Candidate mutations were further restricted to those reachable by single-nucleotide substitutions in the wild-type sequence to simplify synthesis of the resulting combinatorial mutagenesis library. The algorithm begins from an arbitrary starting mutation and iteratively selects further mutations at other residue positions until all residues in the protein have been mutated. The heuristic works by selecting further mutations at each step that maximize the fold and function of the current highest-order mutant combination, that is, the geometric mean of model-predicted AbundancePCA and BindingPCA growth rates. This procedure is then repeated for all possible starting mutations.

To visualize and compare the resulting solutions, we also simulated the median AbundancePCA and BindingPCA growth rates of all candidate combinatorial libraries, calculated using a random sample of 10,000 variants. Although the algorithm is not guaranteed to produce the optimal solution at each Hamming distance from the wild-type sequence, the greedy approach nevertheless achieves solutions in which both phenotypes are predicted to be preserved in variants with more than 30 mutations (Extended Data Fig. 1b ), beyond which one or both phenotypes are lost. Defining viable libraries as those preserving both molecular phenotypes above 70% of the maximal value (that is, the geometric mean of simulated median AbundancePCA and BindingPCA growth rates) resulted in the largest candidate combinatorial library consisting of all combinations of 34 single aa mutations (Fig. 1 and Extended Data Fig. 1b–d ).

Combinatorial library 2

We clustered the contact map (minimal side-chain heavy-atom distance < 5 Å) comprising all GRB2-SH3 surface residues (RSASA ≥ 0.25) existing in secondary structure elements (Extended Data Fig. 4 ) and selected the following four physically proximal residues for saturation combinatorial mutagenesis: H26, M28, A39 and T44 (see Extended Data Fig. 5 ).

Combinatorial library 3

This library was designed to include all combinations of 15 single aa substitutions with mild effects (within one-third of the AbundancePCA fitness interquartile range of the wild type 23 ) in close proximity in the primary sequence and reachable by single-nucleotide substitutions while avoiding mutations in binding interface residues (minimal side-chain heavy-atom distance to the ligand < 5 Å). We used a sliding window approach to determine the number of candidate mutant residues in stretches of 20, 21 and 22 consecutive residues in GRB2-SH3 (Extended Data Fig. 4b ). Only one window with a width of 22 aa (starting at residue position 10) includes 15 candidate positions (Extended Data Fig. 4b ). The final library consisted of all combinations of the following randomly selected candidate mutations at these positions: D10N, P11A, D14N, G15E, G18C, R20S, R21Q, D23E, F24I, H26L, V27I, M28K, D29E, N30T and S31T (see Fig. 4 ).

Combinatorial library 4: SRC

This library was designed using the same greedy algorithm from data and thermodynamic modelling results for SRC 51 , including inferred single aa substitution free energy changes of folding and activity for this protein. The design includes 15 single aa substitutions reachable by single nt substitution in a 22 aa window, located in the N-lobe of the SRC kinase domain, avoiding mutations in the activation loop, subsetting folding and activity ddGs to confident energies (95% confidence interval < 1 kcal mol −1 ) and associated with singles observed in at least seven backgrounds. The final library consisted of all combinations of the following randomly selected candidate mutations at these positions: V329G, G344S, F349V, K343M, E331K, V337A, E332A, M341K, S330N, I336L, T338S, S345T, L346V, P333T and Y340S (see Fig. 6 ).

Mutagenesis library construction and selection assays

Media and buffers used.

LB: 10 g l −1 bacto-tryptone, 5 g l −1 yeast extract, 10 g l −1 NaCl. Autoclaved 20 min at 120 °C.

YPDA: 20 g l −1 glucose, 20 g l −1 peptone, 10 g l −1 yeast extract, 40 mg l −1 adenine sulphate. Autoclaved 20 min at 120 °C.

SORB: 1 M sorbitol, 100 mM LiOAc, 10 mM Tris pH 8.0, 1 mM EDTA. Filter sterilized (0.2-mm nylon membrane, Thermo Scientific).

Plate mixture: 40% PEG3350, 100 mM LiOAc, 10 mM Tris-HCl pH 8.0, 1 mM EDTA pH 8.0. Filter sterilized.

Recovery medium: YPD (20 g l −1 glucose, 20 g l −1 peptone, 10 g l −1 yeast extract) + 0.5 M sorbitol. Filter sterilized.

SC-URA: 6.7 g l −1 yeast nitrogen base without aa, 20 g l −1 glucose, 0.77 g l −1 complete supplement mixture drop-out without uracil. Filter sterilized.

SC-URA/MET/ADE: 6.7 g l −1 yeast nitrogen base without aa, 20 g l −1 glucose, 0.74 g l −1 complete supplement mixture drop-out without uracil, adenine and methionine. Filter sterilized.

Competition medium: SC-URA/MET/ADE + 200 μg ml −1 methotrexate (Merck Life Science), 2% DMSO.

DNA extraction buffer: 2% Triton-X, 1% SDS, 100 mM NaCl, 10 mM Tris-HCl pH 8.0, 1 mM EDTA pH 8.0.

Plasmid construction

For libraries 1–3: GRB2 mutagenesis plasmid pGJJ286: wild-type GRB2-SH3 was digested from pGJJ046 (described previously 23 ) with the restriction enzymes AvrII and HindIII and cloned into the digested plasmid pGJJ191 (described previously 24 ) using T4 ligase (New England Biolabs). AbundancePCA pGJJ046 and pGJJ045 plasmids and BindingPCA pGJJ034 and pGJJ001 plasmids were previously described 23 . For library 4: pTB043 plasmid containing full-length SRC was described previously 51 . pTB043 is based on the same backbone as the AbundancePCA plasmids. The difference is that full-length SRC is fused to the DHFR[3] fragment at its N terminus and to the DHFR[1,2] fragment at its C terminus, so DHFR is reconstituted following correct folding of SRC, whereas unfolded SRC genotypes result in degradation of the fusion protein.

Libraries construction

Libraries 1–3: libraries were constructed in two steps. First, an IDT primer containing the chosen combination of mutations was assembled by Gibson into the mutagenesis plasmid pGJJ286. Libraries were then cloned into the yeast plasmids AbundancePCA pGJJ045 and BindingPCA pGJJ001 by digestion/ligation. For the first step, the libraries into the mutagenesis plasmid were assembled by Gibson reaction (in-house preparation) of two fragments. The vector fragment was obtained by polymerase chain reaction (PCR) amplification of pGJJ286 with the oligos shown in Supplementary Tables 1 and 2 , incubated with DpnI to remove the template and gel purified using QIAquick gel extraction kit (Qiagen). The insert fragment was obtained by mixing equimolar amounts of IDT mutation primer (Supplementary Tables 1 and 2 ) and a reverse elongation primer (Supplementary Tables 1 and 2 ) and incubating for one cycle of annealing/extension with Q5 polymerase (New England Biolabs). dsDNA product was then incubated with ExoSAP-IT (Applied Biosystems) to remove the remaining ssDNA and purified with MinElute columns (Qiagen). 100 ng of vector in a molar ratio of 1:5 with the insert was incubated for 3 h at 50 °C with a Gibson mix 2× prepared in-house. The reaction was desalted by dialysis with membrane filters (MF-Millipore) for 1 h and concentrated 4× using a SpeedVac concentrator (Thermo Scientific). DNA was then transformed into NEB 10-beta High Efficiency Electrocompetent E. coli . Cells were allowed to recover in SOC medium (NEB 10-beta Stable Outgrowth Medium) for 30 min and later transferred to LB medium with spectinomycin overnight. A fraction of cells was also plated into spectinomycin + LB + agar plates to estimate the total number of transformants. 100 ml of each saturated E. coli culture was collected the next morning to extract the mutagenesis plasmid library using the QIAfilter Plasmid Midi Kit (QIAGEN). To obtain the final libraries into the yeast plasmids, libraries in pGJJ286 plasmid were digested with NheI and HindIII, gel purified (MinElute Gel Extraction Kit, QIAGEN) and cloned into pGJJ045 or pGJJ034 digested plasmids with T4 ligase (New England Biolabs) by temperature-cycle ligation following the manufacturer’s instructions, 67 fmol of backbone and 200 fmol of insert in a 33.3-μl reaction. The ligation was desalted by dialysis using membrane filters for 1 h, concentrated 4× using a SpeedVac concentrator (Thermo Scientific) and transformed into NEB 10-beta High Efficiency Electrocompetent E. coli cells.

Library 4: this library was constructed in one step by Gibson reaction of two fragments. The vector fragment was obtained by amplification of pTB043 plasmid with the oligos shown in the Supplementary Tables 1 and 2 . The second fragment was obtained with ten cycles of PCR using mutated IDT primer as template (Supplementary Tables 1 and 2 ).

Methotrexate yeast selection assay

The yeast selection assay was previously described 23 . The high-efficiency yeast transformation protocol described below (adjusted to a pre-culture of 200 ml of YPDA) was scaled up or down, depending on the number of transformants for each library (Supplementary Table 2 ). Three independent pre-cultures of BY4742 were grown in 20 ml of standard YPDA at 30 °C overnight. The next morning, the cultures were diluted into 200 ml of pre-warmed YPDA at an OD 600nm = 0.3 and incubated at 30 °C for 4 h. Cells were then collected and centrifuged for 5 min at 3,000 g , washed with sterile water and SORB medium, resuspended in 8.6 ml of SORB and incubated at room temperature for 30 min. After incubation, 175 μl of 10 mg ml −1 boiled salmon sperm DNA (Agilent Genomics) and 3.5 μg of plasmid library were added to each tube of cells and mixed gently. 35 ml of plate mixture was added to each tube to be incubated at room temperature for a further 30 min. 3.5 ml of DMSO was added to each tube and the cells were then heat shocked at 42 °C for 20 min (inverting tubes from time to time to ensure homogenous heat transfer). After heat shock, cells were centrifuged and resuspended in approximately 50 ml of recovery media and allowed to recover for 1 h at 30 °C. Cells were then centrifuged, washed with SC-URA medium and resuspended in 200 ml SC-URA. 10 μl was plated on SC-URA Petri dishes and incubated for about 48 h at 30 °C to measure the transformation efficiency. The independent liquid cultures were grown at 30 °C for about 48 h until saturation. Saturated cells were diluted again to OD 600nm = 0.1 in SC-URA/MET/ADE media and allowed to grow four generations until OD 600nm = 1.6 at 30 °C and 200 rpm. A fraction of the culture was then used to inoculate 200 ml of competition media containing methotrexate at a starting OD 600nm = 0.05 and the rest was collected and pellets frozen and stored as input. Cells in competition media were allowed to grow for 3–5 generations (Supplementary Table 2 ), collected and frozen and stored as output.

DNA extractions and plasmid quantification

The DNA extraction protocol used was previously described 23 . The protocol below is for 100 ml of collected culture at OD 600nm ≈ 1.6. Protocols were scaled up or down, depending on the library (Supplementary Table 2 ). Cell pellets (one for each experiment input/output replicate) were resuspended in 1 ml of DNA extraction buffer, frozen by dry ice/ethanol bath and incubated at 62 °C in a water bath twice. Subsequently, 1 ml of phenol/chloro/isoamyl in a ratio of 25:24:1 (equilibrated in 10 mM Tris-HCl, 1 mM EDTA, pH 8.0) was added, together with 1 g of acid-washed glass beads (Sigma Aldrich) and the samples were vortexed for 10 min. Samples were centrifuged at room temperature for 30 min at 4,000 rpm and the aqueous phase was transferred into new tubes. The same step was repeated twice. 0.1 ml of NaOAc 3 M and 2.2 ml of pre-chilled absolute ethanol were added to the aqueous phase. The samples were gently mixed and incubated at −20 °C for at least 30 min. After that, they were centrifuged for 30 min at full speed at 4 °C to precipitate the DNA. The ethanol was removed and the DNA pellet was allowed to dry overnight at room temperature. DNA pellets were resuspended in 0.6 ml TE 1X and treated with 5 μl of RNase A (10 mg ml, Thermo Scientific) for 30 min at 37 °C. To desalt and concentrate the DNA solutions, the QIAEX II Gel Extraction Kit was used (50 µl of QIAEX II beads, QIAGEN). The samples were washed twice with PE buffer and eluted twice by 125 µl of 10 mM Tris-HCI buffer, pH 8.5. Finally, plasmid concentrations in the total DNA extract (which also contained yeast genomic DNA) were quantified by quantitative PCR using the primer pair oGJJ152–oGJJ153 that binds to the ori region of the plasmids.

Sequencing library preparation

Libraries 1–3: this was shown in ref. 23 . Briefly, the sequencing libraries were constructed in two consecutive PCR assays. The first PCR (PCR1) was designed to amplify the mutated protein of interest and to increase the nucleotide complexity of the first sequenced bases by introducing frame-shift bases between the adapters and the sequencing region of interest (Supplementary Tables 1 and 2 ). The second PCR (PCR2) was necessary to add the remainder of the Illumina adapter and demultiplexing indexes. PCR2 reactions were run for each sample independently using Hot Start High-Fidelity DNA Polymerase. In this second PCR, the remaining parts of the Illumina adapters were added to the library amplicon. The forward primer (5′ P5 Illumina adapter) was the same for all samples (GJJ_1J), whereas the reverse primer (3′ P7 Illumina adapter) differed by the barcode index (Supplementary Table 3 ) to be subsequently pooled and demultiplexed after deep sequencing. All samples were pooled in an equimolar ratio and gel purified using the QIAEX II Gel Extraction Kit. The purified amplicon library pools were subjected to Illumina 150-bp paired-end NextSeq500 sequencing at the CRG Genomics Core Facility.

Library 4: the method for preparing the library for sequencing was the same as for the other libraries but in the second PCR step, we used a barcoded index in the forward primer as well (5′ P5 Illumina adapter). The purified amplicon library pool was sequenced with an Illumina paired-end NextSeq2000 machine this time.

Sequencing data processing

FastQ files from paired-end sequencing of all AbundancePCA and BindingPCA experiments were processed with DiMSum v1.3 (ref. 52 ) using default settings with small adjustments ( https://github.com/lehner-lab/DiMSum ). Supplementary Table 4 contains DiMSum fitness estimates and associated errors for all experiments. Experimental design files and command-line options required for running DiMSum on these datasets are available on GitHub ( https://github.com/lehner-lab/archstabms ). Variants with fewer than ten input read counts in any replicate were discarded (‘fitnessMinInputCountAll’ option), that is, only variants observed in all replicates above this threshold were retained. For library 1, we also included fitness estimates that derived from a subset of replicates whose input read counts exceeded this threshold (‘fitnessMinInputCountAny’ option; see Fig. 1 ).

For library 1, we also included a wild-type-only sample for sequencing using pGJJ046 as template to derive empirical estimates of sequencing error rates. The FastQ file for this sample was processed identically to those of the replicate input/output samples in the first-pass analysis with DiMSum with permissive base quality thresholds (‘vsearchMinQual = 5’ and ‘vsearchMaxee = 1000’). Read counts for all variants were then adjusted by subtracting the expected number of sequencing errors derived from the wild-type-only sample and proportional to the total sequencing library size of each sample. Finally, fitness estimates and associated errors for library 1 were then obtained from the resulting corrected variant counts with DiMSum (‘countPath’ option).

Thermodynamic modelling with MoCHI

We used MoCHI 43 ( https://github.com/lehner-lab/MoCHI ) to fit all thermodynamic models to combinatorial DMS data using default settings with small adjustments. The software is based on our previously described genotype–phenotype modelling approach 23 , with extra functionality and improvements for ease of use and flexibility 24 , 43 . Models fit to shallow (double-mutant) libraries and used in the analyses described in this work (for example, combinatorial mutagenesis library designs) were obtained using the original software implementation 23 .

We model protein folding as an equilibrium between two states: unfolded (u) and folded (f), and protein binding as an equilibrium between three states: unfolded and unbound (uu), folded and unbound (fu) and folded and bound (fb). We assume that the probability of the unfolded and bound state (ub) is negligible and free energy changes of folding and binding are additive, that is, the total binding and folding free energy changes of an arbitrary variant relative to the wild-type sequence is simply the sum over residue-specific energies corresponding to all constituent single aa substitutions.

We configured MoCHI parameters to specify a neural network architecture consisting of additive trait layers (free energies) for each biophysical trait to be inferred (folding or folding and binding for AbundancePCA or BindingPCA, respectively), as well as one linear transformation layer per observed phenotype. The specified nonlinear transformations ‘TwoStateFractionFolded’ and ‘ThreeStateFractionBound’ derived from the Boltzmann distribution function relate energies to proportions of folded and bound molecules, respectively (see Figs. 2a and 4e,f ). The target (output) data to fit the neural network comprise fitness scores for the wild-type and aa substitution variants of all mutation orders. The inclusion of both first-order and second-order (pairwise energetic coupling) model coefficients in the models was specified using the ‘max_interaction_order’ option.

A random 30% of aa substitution variants of all mutation orders was held out during model training, with 20% representing the validation data and 10% representing the test data. Validation data were used to evaluate training progress and optimize hyperparameters (batch size). Optimal hyperparameters were defined as those resulting in the smallest validation loss after 100 training epochs. Test data were used to assess final model performance.

MoCHI optimizes the parameters θ of the neural network using stochastic gradient descent on a loss function ${\mathcal{L}}[\theta ]$ based on a weighted and regularized form of mean absolute error:

in which y n and σ n are the observed fitness score and associated standard error, respectively, for variant n , ŷ n is the predicted fitness score, N is the batch size and λ 2 is the L 2 regularization penalty. To penalize very large free energy changes (typically associated with extreme fitness scores), we set λ 2 to 10 −6 , representing light regularization. The mean absolute error is weighted by the inverse of the fitness error ( ${\sigma }_{n}^{-1}$ ) to downweight the contribution of less confidently estimated fitness scores to the loss. Furthermore, to capture the uncertainty in fitness estimates, the training data were replaced with a random sample from the fitness error distribution of each variant. The validation and test data were left unaltered.

Models were trained with default settings, that is, for a maximum of 1,000 epochs using the Adam optimization algorithm with an initial learning rate of 0.05 (except for library 1, for which we used an initial learning rate of 0.005). MoCHI reduces the learning rate exponentially ( γ = 0.98) if the validation loss has not improved in the most recent ten epochs compared with the preceding ten epochs. Also, MoCHI stops model training early if the wild-type free energy terms over the most recent ten epochs have stabilized (standard deviation ≤ 10 −3 ).

Free energies are calculated directly from model parameters as follows: Δ G b = θ b RT and Δ G f = θ f RT , in which T = 303 K and R = 0.001987 kcal K −1 mol −1 . We estimated the confidence intervals of model-inferred free energies using a Monte Carlo simulation approach. The variability of inferred free energy changes was calculated between ten separate models fit using data from: (1) independent random training–validation–test splits and (2) independent random samples of fitness estimates from their underlying error distributions. Confident inferred free energy changes are defined as those with Monte Carlo simulation-derived 95% confidence intervals < 1 kcal mol −1 . Supplementary Table 5 contains inferred binding and folding free energy changes and energetic couplings from all second-order models.

Linear model to predict energetic coupling strength

We built a linear model to predict energetic coupling strength (absolute value of energetic coupling terms) from 12 features (see Fig. 3e ), comprising five distance metrics for residue pairs or positions thereof in the protein structure: backbone distance (linear 1D distance separating residue pairs along the primary aa sequence), inter-residue distance (minimal side-chain heavy-atom distance in 3D space), number of core residues (0, 1 or both residues in the pair with RSASA < 0.25), number of binding interface residues (0, 1 or both with minimal side-chain heavy-atom distance to the ligand < 5 Å), number of beta-sheet residues (0, 1 or both in beta strands) and seven features describing the number of chemical bonds or interactions between the atoms of pairs of residues as calculated using the GetContacts software tool ( https://getcontacts.github.io/ ): backbone to backbone hydrogen bonds, side chain to backbone hydrogen bonds, side chain to side chain hydrogen bonds, pi–cation interactions, pi-stacking interactions, salt-bridge interactions and van der Waals interactions. Before running GetContacts, we used PyMOL to fill missing hydrogens (‘h_add’ command), FoldX 53 to restore the wild-type proline at position 54 that is mutated in the reference crystal structure (PDB: 2VWF; ‘PositionScan’ command) and removed GAB2 ligand atoms. The training dataset comprised energetic couplings inferred from library 1 and the test set comprised independently inferred energetic couplings from library 3 (see Fig. 3f ).

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

Data availability

All DNA sequencing data have been deposited in the Gene Expression Omnibus (GEO) with accession number GSE246322 . Associated fitness measurements and free energies are provided in Supplementary Tables 4 and 5 . Shallow double-mutant ddPCA DNA sequencing data for GRB2-SH3 and PSD95-PDZ3 are available in the GEO with accession number GSE184042 , and the processed data used in this study can be found in Supplementary Tables 6 and 7 of the corresponding publication ( https://doi.org/10.1038/s41586-022-04586-4 ). Protein structures are available from the Protein Data Bank for GRB2-SH3 (entry ID: 2VWF ) and SRC (entry ID: 2SRC ).

Code availability

Source code for fitting thermodynamic models (MoCHI) is available at https://github.com/lehner-lab/MoCHI . Source code for all downstream analyses, including DiMSum and MoCHI configuration files, and to reproduce all figures described here is available at https://github.com/lehner-lab/archstabms . An archive of this repository is also publicly available on Zenodo at https://doi.org/10.5281/zenodo.11671164 (ref. 54 ).

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Acknowledgements

This work was funded by a European Research Council Advanced Grant (883742), the Spanish Ministry of Science and Innovation (LCF/PR/HR21/52410004, EMBL Partnership, Severo Ochoa Centre of Excellence), the Bettencourt Schueller Foundation, the AXA Research Fund, Agencia de Gestio d’Ajuts Universitaris i de Recerca (AGAUR, 2017 SGR 1322) and the CERCA Program/Generalitat de Catalunya. A.J.F. was funded by a Ramón y Cajal fellowship (RYC2021-033375-I) financed by the Spanish Ministry of Science and Innovation (MCIN/AEI/10.13039/501100011033) and the European Union (NextGenerationEU/PRTR). A.M.-A. was funded by a fellowship from ”laCaixa” Foundation (ID 100010434, fellowship code B006052). We thank all members of the Lehner Lab for helpful discussions and suggestions.

Author information

Andre J. Faure

Present address: ALLOX, Barcelona, Spain

Jörn M. Schmiedel

Present address: factorize.bio, Berlin, Germany

Authors and Affiliations

Centre for Genomic Regulation (CRG), The Barcelona Institute of Science and Technology, Barcelona, Spain

Andre J. Faure, Aina Martí-Aranda, Cristina Hidalgo-Carcedo, Antoni Beltran, Jörn M. Schmiedel & Ben Lehner

Wellcome Sanger Institute, Wellcome Genome Campus, Hinxton, UK

Aina Martí-Aranda & Ben Lehner

Universitat Pompeu Fabra (UPF), Barcelona, Spain

Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain

You can also search for this author in PubMed Google Scholar

Contributions

B.L. and A.J.F. conceived the project, motivated in part by preliminary unpublished analyses of evolutionary couplings performed by J.M.S. B.L., A.J.F. and A.M.-A. designed the combinatorial mutagenesis libraries and experiments. A.B. provided unpublished data and advice for the SRC experiments. A.M.-M. and C.H.-C. performed the experiments, with help from A.J.F. A.J.F. led the data analysis, with help from A.M.-A. A.J.F. and B.L. wrote the manuscript, with input from A.M.-A. and C.H.-C.

Corresponding authors

Correspondence to Andre J. Faure or Ben Lehner .

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Competing interests.

A.J.F. and B.L. are founders, employees and shareholders of ALLOX. J.M.S. is a founder, employee and shareholder of factorize.bio.

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Nature thanks Willow Coyote-Maestas, Elena Kuzmin and Gabriel Rocklin for their contribution to the peer review of this work. Peer reviewer reports are available.

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Extended data figures and tables

Extended data fig. 1 combinatorial mutagenesis library 1 design and simulations..

a , Violin plot showing distributions of simulated AbundancePCA growth rates (assuming additivity of individual inferred folding free energy changes 23 ) versus number of random aa substitutions ( n = 100,000) for PSD95-PDZ3. Violins are scaled to have the same maximum width. b , Simulated median AbundancePCA/BindingPCA growth rates of optimal combinatorial libraries of increasing maximum aa Hamming distances from the wild type. The horizontal dashed line indicates the 70th percentile of the maximal geometric mean (black). The vertical dashed line indicates the number of aa substitutions selected ( n = 34) for the synthesized combinatorial mutagenesis library 1. c , Violin plot showing simulated distributions of AbundancePCA growth rates versus number of aa substitutions for combinatorial mutagenesis library 1. d , Similar to panel c but showing simulated distributions for BindingPCA growth rates. In panels c and d , the percentage of folded and bound protein variants (predicted fraction folded or bound molecules > 0.5) is shown at each Hamming distance from the wild type.

Extended Data Fig. 2 Residuals versus fitted values for linear and thermodynamic models fit to AbundancePCA data from combinatorial library 1.

a , Residual fitness (observed − predicted) versus predicted fitness for first-order linear models (left) and first-order thermodynamic models (right) evaluated on GRB2-SH3 combinatorial AbundancePCA data (combinatorial library 1; see Fig. 1 ). The smoothed conditional mean (generalized additive model) is shown in red. b , Similar to panel a except models include all first-order and second-order genetic interaction (energetic coupling) terms/coefficients. c , Similar to panel a except results are shown for models that were trained on GRB2-SH3 ddPCA data consisting of single and double aa substitutions only 23 .

Extended Data Fig. 3 Structural determinants of energetic couplings inferred from ddPCA data from combinatorial library 3.

a , Relationship between folding coupling energy strength and minimal inter-residue side-chain heavy-atom distance for combinatorial library 3 (see Fig. 4 ). The mean is shown and error bars indicate 95% confidence intervals from a Monte Carlo simulation approach ( n = 10 experiments). Points are coloured by binned inter-residue distances (see legend in panel b ). Spearman’s ρ is shown for all couplings, as well as those involving pairs of residues separated by more than five residues in the primary sequence. Core residues are indicated as triangles. b , Relationship between folding coupling energy strength and linear sequence (backbone) distance in number of residues. The measure of centre and error bars are as defined in panel a .

Extended Data Fig. 4 Design of combinatorial mutagenesis libraries 2 and 3.

a , Clustered heat map showing structural contacts (minimal side-chain heavy-atom distance < 5 Å) between all GRB2-SH3 surface residues (RSASA ≥ 0.25) existing in secondary structure elements. The four highlighted residues are all physically proximal and were selected as the targets for library 2 saturation combinatorial mutagenesis (see Extended Data Fig. 5 ). b , Bar plot indicating the number of candidate mutant residues in stretches of 20, 21 and 22 consecutive residues in GRB2-SH3 used to design mutagenesis library 3. Candidate mutations were defined as single aa substitutions with mild effects (within one-third of the AbundancePCA fitness interquartile range of the wild type 23 ) in close proximity in the primary sequence and reachable by single-nucleotide substitutions while avoiding mutations in binding interface residues (minimal side-chain heavy-atom distance to the ligand < 5 Å). The selected mutant window size (22 aa residues), residue start position (10) and number of mutated residues (15) is indicated. The final library consisted of all combinations of the following randomly selected candidate mutations at these 15 positions: D10N, P11A, D14N, G15E, G18C, R20S, R21Q, D23E, F24I, H26L, V27I, M28K, D29E, N30T and S31T (see Fig. 4 ).

Extended Data Fig. 5 Saturation combinatorial mutagenesis of a protein surface patch.

a , 3D structure of GRB2-SH3 (PDB: 2VWF) indicating four residues targeted for saturation combinatorial mutagenesis (orange, library 2) and GAB2 ligand (blue). See also Extended Data Fig. 4 . b , Scatter plots showing the reproducibility of fitness estimates from triplicate AbundancePCA experiments. Pearson’s r indicated in red. Rep., biological replicate. c , Histogram showing the number of observed aa variants at increasing Hamming distances from the wild type, in which the x axis is shared with panel d . d , Violin plot showing distributions of AbundancePCA growth rates inferred from deep sequencing data versus number of aa substitutions. The percentage of folded protein variants (predicted fraction folded molecules > 0.5) is shown at each Hamming distance from the wild type. e , Nonlinear relationship (global epistasis) between observed AbundancePCA fitness and changes in free energy of folding. Thermodynamic model fit shown in red. f , Performance of energy model including all first-order and second-order genetic interaction (energetic coupling) terms/coefficients. g , Distributions of folding free energy changes (ΔΔ G , grey) and pairwise energetic couplings (ΔΔΔ G , red). h , Comparisons of the model-inferred single aa substitution free energy changes to previously reported estimates using GRB2-SH3 ddPCA data 23 . Pearson’s r is shown. i , Box plots showing relationship between folding coupling energy strength and minimal inter-residue side-chain heavy-atom distance. Boxes are coloured by inter-residue distance. Spearman’s ρ is shown for all couplings ( n = 2,166 second-order coefficients), as well as the weighted mean per residue pair ( n = 6 residue pairs). j , Relationship between folding coupling energy strength and linear sequence (backbone) distance in number of residues. Boxes are coloured as in panel i . For box plots in panels i and j : centre line, median; box limits, upper and lower quartiles; whiskers, 1.5× interquartile range; n = 2,166 second-order coefficients.

Extended Data Fig. 6 ddPCA data from combinatorial library 3 show that abundant multi-mutants are binding-competent (have conserved fold).

a , Scatter plots showing the reproducibility of fitness estimates from triplicate AbundancePCA experiments for combinatorial library 3 (see Fig. 4 ). Pearson’s r indicated in red. Rep., biological replicate. b , Similar to panel a but showing results from triplicate BindingPCA experiments (same as Fig. 4c ). c , Histogram showing the number of observed aa variants at increasing Hamming distances from the wild type for AbundancePCA, in which the x axis is shared with panel d . d , Violin plot showing distributions of AbundancePCA growth rates inferred from deep sequencing data versus number of aa substitutions. The percentage of folded protein variants (predicted fraction folded molecules > 0.5) is shown at each Hamming distance from the wild type. e , f , Similar to panels c and d but showing results for BindingPCA. The percentage of bound protein variants (predicted fraction folded molecules > 0.5) is shown at each Hamming distance from the wild type in panel f . g , 2D density plots comparing abundance and binding fitness for increasing Hamming distances 1–14 from the wild type as indicated.

Extended Data Fig. 7 Performance of models fit to AbundancePCA data from combinatorial library 3.

a , Performance of first-order two-state thermodynamic model (folded and unfolded states) fit to AbundancePCA data from combinatorial library 3 (see Fig. 4 ). b – d , Performance of first-order ( b ), second-order ( c ) and third-order ( d ) linear models fit to AbundancePCA data from combinatorial library 3 (see Fig. 4 ). R 2 is the proportion of variance explained.

Supplementary information

Reporting summary, peer review file, supplementary table 1.

Primers used in this study

Supplementary Table 2

Description of libraries used in this study

Supplementary Table 3

Illumina indexes used for sequencing

Supplementary Table 4

Fitness estimates for all experiments

Supplementary Table 5

Inferred free energy changes and associated annotations for all experiments

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Sound Waves

What is Sonometer?

A sonometer is defined as

The device that is used for demonstrating the relationship between the frequency of the sound that is produced by the string when it is plucked and the tension, length, and mass per unit length of the string.

The sound is produced in the transverse standing wave in the string.

Sonometer Diagram

Construction of Sonometer

A hollow box of one meter long is used for making the sonometer.
This hollow box consists of a uniform metallic string attached to it.
The first end of the string is connected to the hook while the second end is connected to the weight hanger with the help of a pulley.
The number of strings used is one and therefore, it is also known as the monochord.
In order to increase the tension of the string, weights are added at the free end.
To change the vibrating length of the string which is stretched, two adjustable knives are placed and their positions are adjusted accordingly.

Working of Sonometer

Once the set-up is done (as shown in the sonometer diagram), a transverse standing wave is produced at the edges of the knives with the nodes. Along with the nodes, there is a formation of the anti-nodes.

The mathematical explanation to the working of the sonometer is as follows:

Let l be the length of the vibrating element

Let f be the frequency of the vibrating element. T be the tension created in the string and μ be mass per unit length. Then,

If ⍴ is the density of the material and d is the diameter of the string, the mass per unit length μ is given as

μ = area x density = πr 2 ⍴ = π⍴d 2 /4

Then frequency is

Laws of Transverse Vibrations

The laws of transverse vibrations on a stretched string can be divided into two laws, and they are:

Law of length
Law of tension
Law of mass

Law of Length

The law of length states that the frequency of the vibration of a stretched string and its resonating length varies inversely as long as the mass per unit length and the tension of the string is constant.

Law of Tension

The law of tension states that the frequency of vibration of a stretched string and the square root of its tension varies directly as long as the resonating length and the mass per unit length of the string is constant.

Law of Mass

The law of mass states that the frequency of vibration of a stretched string and the square root of its mass per unit length varies inversely proportional as long as the length and the tension is constant.

What is the use of sonometer?

We already know that the sonometer is used for demonstrating the relationship between the frequency and the tension, length, and mass per unit length of the string. Therefore, the following can be determined using a sonometer:

The frequency of the tuning fork
The frequency of the alternating current
The tension in the string
The unknown mass of the object hanging down

Stay tuned with BYJU’S to learn more about other concepts of Physics.

Frequently Asked Questions – FAQs

What is an example of a standing wave.

An example of a standing wave is noticed in stringed musical instruments.

What is meant by a standing wave?

A standing wave is a combination of two waves moving in opposite directions such that they have the same amplitude and frequency and is obtained due to interference. The standing wave is also known as the stationary wave.

Why are holes kept on one side of the sonometer box?

The holes in the sonometer box act as a way through which the frequency of vibration of the string is communicated inside the hollow portion of the box.

Which material is used for sonometer wire?

The wire that is used in the sonometer is soft iron.

Who invented sonometer?

The first monochord was invented by Pythagoras in the 7th century. Later in the mid 1800, Albert Marloye, a French instrument maker, modified the monochord into a differential sonometer.

Audible and Inaudible Sounds

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Sonometer
12th Physics Experiment No.4 Answers [Science]~Sonometer
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Sonometer Experiment 1 Law of Length
Sonometer Experiment ‒ Objective, Procedure, and Tips
12th science physics experiment no 4 sonometer law of length answers

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PDF Experiment88
4. Measure the vibrating length and note the tension in the string. 5. Increase the load in steps of ½ kg and each time find the vibrating length. 6. Switch off the ac supply. Untie the wire of the sonometer from its peg and find its mass in a physical balance. Calculate mass of 100 cm sonometer wire. Hence find the mass per unit length, m for ...
Find Relation Between The Length Of A Given Wire And Tension For
Check for the frequencies again. The length of the wire AB should be reduced continuously until the frequency of the wire becomes equal to the frequency of the tuning fork. Place a V shaped paper rider R in the middle of the wire. Strike the tuning fork against the rubber pad and touch the lower end of the handle with the sonometer board.
Sonometer Experiment 1 Law of Length
This video covers Famous Experiment Sonometer in which we will verify the laws of vibrating string and in this video, Law of length is verified Experimentall...
Physics practical. 12 sci. Experiment no. 4 Sonometer: Law of Length
Physics practical. 12 sci. Experiment no. 4 Sonometer: Law of Length . Observations, Graph, Result
PDF Apparatus and Material Required Escription of Apparatus
EXPERIMENT15 Fig. E 15.1: A Sonometer (E 15.1) A IM (i) To study the relation between frequency and length of a given wire under constant tension using a sonometer . (ii) To study the relation between the length of a given wire and tension for constant fr equency using a sonometer . A PPARATUS AND MATERIAL REQUIRED Sonometer, six tuning forks
Sonometer Experiment ‒ Objective, Procedure, and Tips
Place the sonometer box on top of your dry lab workbench in your physics practical class. Place it in a way so that the end with the pulley is flush with the open side of the table, so that something can hang from the pulley. Get the wire you have to experiment with and verify that you have enough length of it.
sonometer- Law of length(1).pdf
Set up the sonometer on the table and clean the groove on the pulley to ensure that it has, minimum friction. Stretch the wire by placing a suitable load on the hanger., , 2. Add suitable mass to the hanger to the sonometer wire. Keep it constant throughout the, experiment., , 3. Place a light paper rider on the wire midway between knife edges ...
Relationship between frequency and length of wire under constant
For example, if the length of the string is l, and it is stretched with a tension (T) and has a mass (m) per unit length, then the frequency of the vibration is expressed as, F = 1/2l T/m− −−−√ F = 1 / 2 l T / m . In this expression, F is considered as the constant, and in turn, it states that, m and T/1− −−√ T / 1 is also ...
Class 12 Physics Experimet 4. Sonometer law of length
We study Class 12 Physics Experimet 4. Sonometer law of length. 12 Physics Practical 4. Law of length done with observation with calculation. 12 Practical A...
CBSE Physics Practical Class 12 Lab Manual for 2023-24 Board Exam
CBSE Class 12 Physics Activities Section B. 1. To identify a diode, an LED, a resistor and a capacitor from a mixed collection of such items. 2. Use of a multimeter to see the unidirectional flow of current in the case of a diode and an LED and check whether a given electronic component (e.g., diode) is in working order.
To Study The Relation Between Frequency And Length Of Wire Under
Place the sonometer on the table as shown in the figure. Oil the pulley to make it frictionless. Put suitable maximum weight to the hanger. Move the wooden bridge P outward to include the maximum length of wire AB between them. Pick the tuning fork of the least frequency from the set. Make it vibrate by striking its prong on a rubber pad.
Sonometer
Sonometer Working: The Theory of Standing Waves. We can assume that there are two parts of the sonometer wire: (a) The part of the wire between the knife edges and, (b) The part of the wire other than the part between the knife edges. Only part (a) is involved in a discussion of standing waves. Part (b) is the other supporting part, attached to ...
Sonometer
Sonometer I :-Part 1)https://youtu.be/0nhecozwNz4Part 2)https://youtu.be/LRZqF8iC1M8Part 3)https://youtu.be/oe4EFsZfhM8इतर महत्वाच्या ...
Class 12 Physics practical reading To find the frequency of ...
1. Set up the apparatus as shown in Fig. Setup for finding frequency of A.C. mains by sonometer. 2. Place the horse - shoe magnet in the middle of the wedges. [In the labs, where electromagnet is used instead of a horse-shoe magnet, the electromagnet should be adjusted a little above the sonometer wire so that one of its poles lies close to the ...
Sonometer Experiment
In this article, we shall study construction and working of sonometer, and its use to verify the laws of string. Laws of Vibrating String: Law of Length: If the tension in the string and its mass per unit length of wire remains constant, then the frequency of transverse vibration of a stretched string is inversely proportional to the vibrating ...
To Study the Relation Between the Length of a Given Wire and Tension
To Study the Relation Between the Length of a Given Wire and Tension for Constant Frequency Using Sonometer Physics Lab ManualNCERT Solutions Class 11 Physics Sample Papers Aim To study the relation between the length of a given wire and tension for constant frequency using sonometer. Apparatus A sonometer, a set of eight tuning forks, […]
Experiment No 04 : SONOMETER LAW OF LENGTH / Physics Practical / Class
Experiment No 04 : SONOMETER LAW OF LENGTH / Physics Practical / Class 12th HSC Board #study_tech_boards_mania #physics_practical#12thclassphysics #12thclass...
Sonometer
Description. Sonometer consists of a hollow rectangular wooden box of more than one-meter length, with a hook at one end and a pulley at the other end. One end of a string is fixed at the hook and the other end passes over the pulley. A weight hanger is attached to the free end of the string. Two adjustable wooden bridges are put over the board ...
PDF Experiments 14-17.pmd
Fig. E 15.1: A Sonometer (E 15.1) (i) To study the relation between frequency and length of a given wire under constant tension using a sonometer. (ii) To study the relation between the length of a given wire and tension for constant frequency using a sonometer. Sonometer, six tuning forks of known frequencies, metre scale, rubber pad, paper
The genetic architecture of protein stability
There are more ways to synthesize a 100-amino acid (aa) protein (20100) than there are atoms in the universe. Only a very small fraction of such a vast sequence space can ever be experimentally or ...
Sonometer Law of Length 12th Physics Lab Practical Experiment ...
Telegram Link-https://t.me/professorofscience/1Join Telegram for VivaQuestions,Numericals and Important QuestionsLike, Share and Subscribe to this channel to...
To Find The Frequency Of AC Mains With A Sonometer
Measure the length and record it in length increasing column. Now take a tuning fork of minimum known frequency and adjust the wire length with the vibrating tuning fork. Repeat step 11 above with tuning forks of other known frequencies.
Sonometer: Definition, Diagram, Construction, Working, Uses
A sonometer is defined as. The device that is used for demonstrating the relationship between the frequency of the sound that is produced by the string when it is plucked and the tension, length, and mass per unit length of the string. The sound is produced in the transverse standing wave in the string.
Sonometer 2
This video covers Famous Experiment Sonometer in which we will verify Law of Tension. This is part of syllabus of class 12 students from physics background. ...

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To Study the Relation Between the Length of a Given Wire and Tension for Constant Frequency Using Sonometer

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The genetic architecture of protein stability

Sampling a 10 10 sequence space

Genetic prediction with energy models

Couplings improve genetic prediction

Physical contacts and backbone proximity

Couplings decay along the peptide chain

Higher-order mutants fold and function

Multi-phenotype genetic prediction

Allostery in ligand-proximal residues

SRC kinase combinatorial mutagenesis

Combinatorial mutagenesis library designs

Combinatorial library 2

Combinatorial library 3

Combinatorial library 4: SRC

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Methotrexate yeast selection assay

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Extended data figures and tables

Extended Data Fig. 2 Residuals versus fitted values for linear and thermodynamic models fit to AbundancePCA data from combinatorial library 1.

Extended Data Fig. 3 Structural determinants of energetic couplings inferred from ddPCA data from combinatorial library 3.

Extended Data Fig. 4 Design of combinatorial mutagenesis libraries 2 and 3.

Extended Data Fig. 5 Saturation combinatorial mutagenesis of a protein surface patch.

Extended Data Fig. 6 ddPCA data from combinatorial library 3 show that abundant multi-mutants are binding-competent (have conserved fold).